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torch algorithm utils

lm_opt 

lm_opt(f, theta0, ytrue, batch_dims=0, iters=10, residtol=None, loss_mult=1, loss_shift=0, f_kwargs_vec={}, f_kwargs_no_vec={}, lam0=1e-06, alpha0=1.0, lam_factors=[[1 / 2, 1, 2]], alpha_factors=[[1 / 2, 1, 2]], vmap_chunk_size=None, jacmode='auto', verbose=False, verbose_indent=4, quantiles_losses=[0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100], quantiles_lams=[0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100], quantiles_alphas=[0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100], verbose_quantiles_losses=[5, 25, 50, 75, 90], verbose_quantiles_lams=[5, 25, 50, 75, 90], verbose_quantiles_alphas=[5, 25, 50, 75, 90], verbose_times=True, warn=True, store_data_iters=False, store_all_data=False)

Levenberg--Marquardt optimization

Parameters:

Name Type Description Default
f func

Residual function.

required
theta0 Tensor

Initial guess for parameters \(\theta\).

required
ytrue Tensor

True y values, i.e. f(theta_true).

required
batch_dims int

Number of batch dimension.

0
iters int

Number of iterations.

10
residtol float

Non-negative tolerance on the maximum residual for early stopping, defaults to 1e-12 for torch.float64 and 2.5e-4 for torch.float32.

 None
loss_mult bool

Scalar amount by which to multiply the loss so loss = loss_mult*torch.sum(resid**2,dim=-1)+loss_shift.

 1
loss_shift bool

Scalar amount by which to shift the loss so loss = loss_mult*torch.sum(resid**2,dim=-1)+loss_shift.

 0
f_kwargs_vec dict

Keyword arguments to f which will be vectorized over the first dimension.

{}
f_kwargs_no_vec dict

Keyword arguments to f which will not be vectorized over the first dimension.

{}
lam0 float

Initial positive relaxation parameter \(\lambda\).

 1e-06
alpha0 float

Initial positive step size \(\alpha\).

 1.0
lam_factors Tensor

Either a 1D torch.Tensor or list of 1d torch.Tensor.

  • Passing in a float for lam_factors is equivalent to passing in torch.tensor([lam_factors]) on the correct device
  • If a 1D torch.Tensor for lam_factors will consider all lam*lam_factors options at each step.
  • If a list of 1D torch.Tensors are passed in for lam_factors, iterations will cycle through the list and then return to the start after exhausting the list.
 [[1 / 2, 1, 2]]
alpha_factors Tensor

Either a 1D torch.Tensor or list of 1d torch.Tensor.

  • Passing in a float for alpha_factors is equivalent to passing in torch.tensor([alpha_factors]) on the correct device
  • If a 1D torch.Tensor for alpha_factors will consider all lam*alpha_factors options at each step.
  • If a list of 1D torch.Tensors are passed in for alpha_factors, iterations will cycle through the list and then return to the start after exhausting the list.
[[1 / 2, 1, 2]]
vmap_chunk_size int

Parameter chunksize to pass to torch.vmap.

 None
jacmode bool

Choose between torch.func.jacfwd and torch.func.jacrev using options:

  • "fwd": Use torch.func.jacfwd.
  • "rev": Use torch.func.jacrev.
  • "auto": Choose between torch.func.jacfwd and torch.func.jacrev depending on the size of the inputs and outputs.
'auto'
verbose int

Controls logging verbosity

  • If True, perform logging.
  • If a positive int, only log every verbose iterations.
  • If None, set to a reasonable positive int based on the maximum number of iterations
  • If False, don't log.
False
verbose_indent int

Non-negative number of indentation spaces for logging.

 4
quantiles_losses list

Loss quantiles to record.

 [0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100]
quantiles_lams list

\(\lambda\) quantiles to record.

 [0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100]
quantiles_alphas list

\(\alpha\) quantiles to record.

 [0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100]
verbose_quantiles_losses list

Loss quantiles to show in verbose log.

 [5, 25, 50, 75, 90]
verbose_quantiles_lams list

\(\lambda\) quantiles to show in verbose log.

 [5, 25, 50, 75, 90]
verbose_quantiles_alphas list

\(\alpha\) quantiles to show in verbose log.

 [5, 25, 50, 75, 90]
verbose_times bool

If False, do not show the times in the verbose log. This is mostly for testing where timing is not reproducible.

True
warn bool

If False, then suppress warnings.

 True
store_data_iters int

Controls storage iterations with the same options as verbose. If store_data_iters==0, then the data is not collected or returned.

  • If True, store every iteration.
  • If a positive int, only store every store_data_iters iterations.
  • If None, set to a reasonable positive int based on the maximum number of iterations
  • If False, don't store data, and do not return data
False
store_all_data bool

If True, store the x values as well as the metrics.

False

Returns:

Name Type Description
theta Tensor

Optimized parameters.
data dict

Iteration data, only returned when store_data_iters>0

Examples:

>>> torch.set_default_dtype(torch.float64)
>>> rng = torch.Generator().manual_seed(7)

Standard example

>>> x = torch.rand((10,4,),generator=rng)
>>> theta_true = torch.rand((4,),generator=rng)
>>> ytrue = torch.exp((x*theta_true).sum(-1)) # (10,)
>>> def f(theta):
...     yhat = torch.exp((x*theta[...,None,:]).sum(-1)) # (...,10)
...     return yhat
>>> theta_hat,data = lm_opt(
...     f = f, 
...     theta0 = torch.rand_like(theta_true,generator=rng),
...     ytrue = ytrue,
...     iters = 3,
...     batch_dims = 0,
...     verbose = True,
...     verbose_times = False,
...     store_data_iters = None,
...     store_all_data = True,
...     )
    iter i     | losses_quantiles                                          | lams_quantiles                                            | alphas_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 2.3e+01   | 2.3e+01   | 2.3e+01   | 2.3e+01   | 2.3e+01   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 3.4e+00   | 3.4e+00   | 3.4e+00   | 3.4e+00   | 3.4e+00   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   
    2          | 4.5e-02   | 4.5e-02   | 4.5e-02   | 4.5e-02   | 4.5e-02   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   
    3          | 4.8e-06   | 4.8e-06   | 4.8e-06   | 4.8e-06   | 4.8e-06   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   
>>> torch.allclose(theta_hat,theta_true,atol=5e-2)
True
>>> print_data_signatures(data)
    data['theta'].shape = (4,)
    data['iterrange'].shape = (4,)
    data['times'].shape = (4,)
    data['losses_quantiles']
        data['losses_quantiles']['0'].shape = (4,)
        data['losses_quantiles']['1'].shape = (4,)
        data['losses_quantiles']['5'].shape = (4,)
        data['losses_quantiles']['10'].shape = (4,)
        data['losses_quantiles']['25'].shape = (4,)
        data['losses_quantiles']['40'].shape = (4,)
        data['losses_quantiles']['50'].shape = (4,)
        data['losses_quantiles']['60'].shape = (4,)
        data['losses_quantiles']['75'].shape = (4,)
        data['losses_quantiles']['90'].shape = (4,)
        data['losses_quantiles']['95'].shape = (4,)
        data['losses_quantiles']['99'].shape = (4,)
        data['losses_quantiles']['100'].shape = (4,)
    data['lams_quantiles']
        data['lams_quantiles']['0'].shape = (4,)
        data['lams_quantiles']['1'].shape = (4,)
        data['lams_quantiles']['5'].shape = (4,)
        data['lams_quantiles']['10'].shape = (4,)
        data['lams_quantiles']['25'].shape = (4,)
        data['lams_quantiles']['40'].shape = (4,)
        data['lams_quantiles']['50'].shape = (4,)
        data['lams_quantiles']['60'].shape = (4,)
        data['lams_quantiles']['75'].shape = (4,)
        data['lams_quantiles']['90'].shape = (4,)
        data['lams_quantiles']['95'].shape = (4,)
        data['lams_quantiles']['99'].shape = (4,)
        data['lams_quantiles']['100'].shape = (4,)
    data['alphas_quantiles']
        data['alphas_quantiles']['0'].shape = (4,)
        data['alphas_quantiles']['1'].shape = (4,)
        data['alphas_quantiles']['5'].shape = (4,)
        data['alphas_quantiles']['10'].shape = (4,)
        data['alphas_quantiles']['25'].shape = (4,)
        data['alphas_quantiles']['40'].shape = (4,)
        data['alphas_quantiles']['50'].shape = (4,)
        data['alphas_quantiles']['60'].shape = (4,)
        data['alphas_quantiles']['75'].shape = (4,)
        data['alphas_quantiles']['90'].shape = (4,)
        data['alphas_quantiles']['95'].shape = (4,)
        data['alphas_quantiles']['99'].shape = (4,)
        data['alphas_quantiles']['100'].shape = (4,)
    data['thetas'].shape = (4, 4)
    data['losses'].shape = (4,)
    data['lams'].shape = (4,)
    data['alphas'].shape = (4,)

Batched example 

>>> x = torch.rand((3,3,3,2,2),generator=rng)
>>> theta_true = torch.rand((4,4,2,2),generator=rng)
>>> ytrue = torch.exp((x*theta_true[...,None,None,None,:,:]).sum((-2,-1))) # (4,4,3,3,3)
>>> def f(theta):
...     yhat = torch.exp((x*theta[...,None,None,None,:,:]).sum((-2,-1))) # (...,3,3,3)
...     return yhat
>>> theta_hat,data = lm_opt(
...     f = f, 
...     theta0 = torch.rand_like(theta_true,generator=rng),
...     ytrue = ytrue,
...     iters = 2,
...     batch_dims = 2,
...     lam_factors = [torch.tensor([1/4,1/2,1,2,4])],
...     alpha_factors = [torch.tensor([2/3,1,3/2])],
...     verbose = True,
...     verbose_times = False,
...     store_data_iters = None,
...     store_all_data = True,
...     )
    iter i     | losses_quantiles                                          | lams_quantiles                                            | alphas_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 5.5e+00   | 2.5e+01   | 5.3e+01   | 1.3e+02   | 5.1e+02   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 7.4e-02   | 6.4e-01   | 1.1e+00   | 2.2e+00   | 7.7e+01   | 2.5e-07   | 2.5e-07   | 2.5e-07   | 2.5e-07   | 4.0e-06   | 6.7e-01   | 6.7e-01   | 1.0e+00   | 1.1e+00   | 1.5e+00   
    2          | 1.0e-05   | 6.9e-04   | 1.8e-03   | 3.9e-03   | 1.2e+00   | 6.2e-08   | 6.2e-08   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 6.7e-01   | 6.7e-01   | 1.0e+00   | 1.1e+00   | 1.5e+00   
>>> torch.allclose(theta_hat,theta_true,atol=5e-2)
False
>>> print_data_signatures(data)
    data['theta'].shape = (4, 4, 2, 2)
    data['iterrange'].shape = (3,)
    data['times'].shape = (3,)
    data['losses_quantiles']
        data['losses_quantiles']['0'].shape = (3,)
        data['losses_quantiles']['1'].shape = (3,)
        data['losses_quantiles']['5'].shape = (3,)
        data['losses_quantiles']['10'].shape = (3,)
        data['losses_quantiles']['25'].shape = (3,)
        data['losses_quantiles']['40'].shape = (3,)
        data['losses_quantiles']['50'].shape = (3,)
        data['losses_quantiles']['60'].shape = (3,)
        data['losses_quantiles']['75'].shape = (3,)
        data['losses_quantiles']['90'].shape = (3,)
        data['losses_quantiles']['95'].shape = (3,)
        data['losses_quantiles']['99'].shape = (3,)
        data['losses_quantiles']['100'].shape = (3,)
    data['lams_quantiles']
        data['lams_quantiles']['0'].shape = (3,)
        data['lams_quantiles']['1'].shape = (3,)
        data['lams_quantiles']['5'].shape = (3,)
        data['lams_quantiles']['10'].shape = (3,)
        data['lams_quantiles']['25'].shape = (3,)
        data['lams_quantiles']['40'].shape = (3,)
        data['lams_quantiles']['50'].shape = (3,)
        data['lams_quantiles']['60'].shape = (3,)
        data['lams_quantiles']['75'].shape = (3,)
        data['lams_quantiles']['90'].shape = (3,)
        data['lams_quantiles']['95'].shape = (3,)
        data['lams_quantiles']['99'].shape = (3,)
        data['lams_quantiles']['100'].shape = (3,)
    data['alphas_quantiles']
        data['alphas_quantiles']['0'].shape = (3,)
        data['alphas_quantiles']['1'].shape = (3,)
        data['alphas_quantiles']['5'].shape = (3,)
        data['alphas_quantiles']['10'].shape = (3,)
        data['alphas_quantiles']['25'].shape = (3,)
        data['alphas_quantiles']['40'].shape = (3,)
        data['alphas_quantiles']['50'].shape = (3,)
        data['alphas_quantiles']['60'].shape = (3,)
        data['alphas_quantiles']['75'].shape = (3,)
        data['alphas_quantiles']['90'].shape = (3,)
        data['alphas_quantiles']['95'].shape = (3,)
        data['alphas_quantiles']['99'].shape = (3,)
        data['alphas_quantiles']['100'].shape = (3,)
    data['thetas'].shape = (3, 4, 4, 2, 2)
    data['losses'].shape = (3, 4, 4)
    data['lams'].shape = (3, 4, 4)
    data['alphas'].shape = (3, 4, 4)
Source code in agsutil/algos.py 
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def lm_opt(
        f,
        theta0,
        ytrue,
        batch_dims = 0, 
        iters = 10,
        residtol = None,
        loss_mult = 1,
        loss_shift = 0,
        f_kwargs_vec = {},
        f_kwargs_no_vec = {},
        lam0 = 1e-6,
        alpha0 = 1e0,
        lam_factors = [[1/2,1,2]],
        alpha_factors = [[1/2,1,2]],
        vmap_chunk_size = None,
        jacmode = "auto",
        verbose = False, 
        verbose_indent = 4,
        quantiles_losses = [0,1,5,10,25,40,50,60,75,90,95,99,100],
        quantiles_lams =   [0,1,5,10,25,40,50,60,75,90,95,99,100],
        quantiles_alphas = [0,1,5,10,25,40,50,60,75,90,95,99,100],
        verbose_quantiles_losses = [5,25,50,75,90],
        verbose_quantiles_lams =   [5,25,50,75,90],
        verbose_quantiles_alphas = [5,25,50,75,90],
        verbose_times = True, 
        warn = True,
        store_data_iters = False,
        store_all_data = False, 
        ):
    r"""
    Levenberg--Marquardt optimization 

    Args:
        f (func): Residual function. 
        theta0 (torch.Tensor): Initial guess for parameters $\theta$. 
        ytrue (torch.Tensor): True `y` values, i.e. `f(theta_true)`. 
        batch_dims (int): Number of batch dimension. 
        iters (int): Number of iterations. 
        residtol (float): Non-negative tolerance on the maximum residual for early stopping, defaults to `1e-12` for `torch.float64` and `2.5e-4` for `torch.float32`.
        loss_mult (bool): Scalar amount by which to multiply the loss so `loss = loss_mult*torch.sum(resid**2,dim=-1)+loss_shift`.
        loss_shift (bool): Scalar amount by which to shift the loss so `loss = loss_mult*torch.sum(resid**2,dim=-1)+loss_shift`.
        f_kwargs_vec (dict): Keyword arguments to `f` which will be vectorized over the first dimension. 
        f_kwargs_no_vec (dict): Keyword arguments to `f` which will not be vectorized over the first dimension. 
        lam0 (float): Initial positive relaxation parameter $\lambda$.
        alpha0 (float): Initial positive step size $\alpha$.
        lam_factors (torch.Tensor): Either a 1D `torch.Tensor` or list of 1d `torch.Tensor`. 

            - Passing in a `float` for `lam_factors` is equivalent to passing in `torch.tensor([lam_factors])` on the correct device
            - If a 1D `torch.Tensor` for `lam_factors` will consider all `lam*lam_factors` options at each step. 
            - If a list of 1D `torch.Tensor`s are passed in for `lam_factors`, iterations will cycle through the list and then return to the start after exhausting the list.

        alpha_factors (torch.Tensor): Either a 1D `torch.Tensor` or list of 1d `torch.Tensor`. 

            - Passing in a `float` for `alpha_factors` is equivalent to passing in `torch.tensor([alpha_factors])` on the correct device
            - If a 1D `torch.Tensor` for `alpha_factors` will consider all `lam*alpha_factors` options at each step. 
            - If a list of 1D `torch.Tensor`s are passed in for `alpha_factors`, iterations will cycle through the list and then return to the start after exhausting the list. 

        vmap_chunk_size (int): Parameter `chunksize` to pass to `torch.vmap`.
        jacmode (bool): Choose between `torch.func.jacfwd` and `torch.func.jacrev` using options:

            - `"fwd"`: Use `torch.func.jacfwd`.
            - `"rev"`: Use `torch.func.jacrev`.
            - `"auto"`: Choose between `torch.func.jacfwd` and `torch.func.jacrev` depending on the size of the inputs and outputs. 

        verbose (int): Controls logging verbosity

            - If `True`, perform logging. 
            - If a positive int, only log every verbose iterations. 
            - If `None`, set to a reasonable positive int based on the maximum number of iterations
            - If `False`, don't log. 

        verbose_indent (int): Non-negative number of indentation spaces for logging.
        quantiles_losses (list): Loss quantiles to record.
        quantiles_lams (list): $\lambda$ quantiles to record.
        quantiles_alphas (list): $\alpha$ quantiles to record.
        verbose_quantiles_losses (list): Loss quantiles to show in verbose log.
        verbose_quantiles_lams (list): $\lambda$ quantiles to show in verbose log.
        verbose_quantiles_alphas (list): $\alpha$ quantiles to show in verbose log.
        verbose_times (bool): If `False`, do not show the times in the verbose log. This is mostly for testing where timing is not reproducible. 
        warn (bool): If `False`, then suppress warnings.
        store_data_iters (int): Controls storage iterations with the same options as verbose. If `store_data_iters==0`, then the data is not collected or returned. 

            - If `True`, store every iteration. 
            - If a positive int, only store every `store_data_iters` iterations. 
            - If `None`, set to a reasonable positive int based on the maximum number of iterations
            - If `False`, don't store data, and do not return data 

        store_all_data (bool): If `True`, store the `x` values as well as the metrics. 

    Returns:
        theta (torch.Tensor): Optimized parameters.
        data (dict): Iteration data, only returned when `store_data_iters>0`

    Examples:

        >>> torch.set_default_dtype(torch.float64)
        >>> rng = torch.Generator().manual_seed(7)


    Standard example

        >>> x = torch.rand((10,4,),generator=rng)
        >>> theta_true = torch.rand((4,),generator=rng)
        >>> ytrue = torch.exp((x*theta_true).sum(-1)) # (10,)
        >>> def f(theta):
        ...     yhat = torch.exp((x*theta[...,None,:]).sum(-1)) # (...,10)
        ...     return yhat
        >>> theta_hat,data = lm_opt(
        ...     f = f, 
        ...     theta0 = torch.rand_like(theta_true,generator=rng),
        ...     ytrue = ytrue,
        ...     iters = 3,
        ...     batch_dims = 0,
        ...     verbose = True,
        ...     verbose_times = False,
        ...     store_data_iters = None,
        ...     store_all_data = True,
        ...     )
            iter i     | losses_quantiles                                          | lams_quantiles                                            | alphas_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 2.3e+01   | 2.3e+01   | 2.3e+01   | 2.3e+01   | 2.3e+01   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 3.4e+00   | 3.4e+00   | 3.4e+00   | 3.4e+00   | 3.4e+00   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   
            2          | 4.5e-02   | 4.5e-02   | 4.5e-02   | 4.5e-02   | 4.5e-02   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   
            3          | 4.8e-06   | 4.8e-06   | 4.8e-06   | 4.8e-06   | 4.8e-06   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-07   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   | 5.0e-01   
        >>> torch.allclose(theta_hat,theta_true,atol=5e-2)
        True
        >>> print_data_signatures(data)
            data['theta'].shape = (4,)
            data['iterrange'].shape = (4,)
            data['times'].shape = (4,)
            data['losses_quantiles']
                data['losses_quantiles']['0'].shape = (4,)
                data['losses_quantiles']['1'].shape = (4,)
                data['losses_quantiles']['5'].shape = (4,)
                data['losses_quantiles']['10'].shape = (4,)
                data['losses_quantiles']['25'].shape = (4,)
                data['losses_quantiles']['40'].shape = (4,)
                data['losses_quantiles']['50'].shape = (4,)
                data['losses_quantiles']['60'].shape = (4,)
                data['losses_quantiles']['75'].shape = (4,)
                data['losses_quantiles']['90'].shape = (4,)
                data['losses_quantiles']['95'].shape = (4,)
                data['losses_quantiles']['99'].shape = (4,)
                data['losses_quantiles']['100'].shape = (4,)
            data['lams_quantiles']
                data['lams_quantiles']['0'].shape = (4,)
                data['lams_quantiles']['1'].shape = (4,)
                data['lams_quantiles']['5'].shape = (4,)
                data['lams_quantiles']['10'].shape = (4,)
                data['lams_quantiles']['25'].shape = (4,)
                data['lams_quantiles']['40'].shape = (4,)
                data['lams_quantiles']['50'].shape = (4,)
                data['lams_quantiles']['60'].shape = (4,)
                data['lams_quantiles']['75'].shape = (4,)
                data['lams_quantiles']['90'].shape = (4,)
                data['lams_quantiles']['95'].shape = (4,)
                data['lams_quantiles']['99'].shape = (4,)
                data['lams_quantiles']['100'].shape = (4,)
            data['alphas_quantiles']
                data['alphas_quantiles']['0'].shape = (4,)
                data['alphas_quantiles']['1'].shape = (4,)
                data['alphas_quantiles']['5'].shape = (4,)
                data['alphas_quantiles']['10'].shape = (4,)
                data['alphas_quantiles']['25'].shape = (4,)
                data['alphas_quantiles']['40'].shape = (4,)
                data['alphas_quantiles']['50'].shape = (4,)
                data['alphas_quantiles']['60'].shape = (4,)
                data['alphas_quantiles']['75'].shape = (4,)
                data['alphas_quantiles']['90'].shape = (4,)
                data['alphas_quantiles']['95'].shape = (4,)
                data['alphas_quantiles']['99'].shape = (4,)
                data['alphas_quantiles']['100'].shape = (4,)
            data['thetas'].shape = (4, 4)
            data['losses'].shape = (4,)
            data['lams'].shape = (4,)
            data['alphas'].shape = (4,)

    Batched example 

        >>> x = torch.rand((3,3,3,2,2),generator=rng)
        >>> theta_true = torch.rand((4,4,2,2),generator=rng)
        >>> ytrue = torch.exp((x*theta_true[...,None,None,None,:,:]).sum((-2,-1))) # (4,4,3,3,3)
        >>> def f(theta):
        ...     yhat = torch.exp((x*theta[...,None,None,None,:,:]).sum((-2,-1))) # (...,3,3,3)
        ...     return yhat
        >>> theta_hat,data = lm_opt(
        ...     f = f, 
        ...     theta0 = torch.rand_like(theta_true,generator=rng),
        ...     ytrue = ytrue,
        ...     iters = 2,
        ...     batch_dims = 2,
        ...     lam_factors = [torch.tensor([1/4,1/2,1,2,4])],
        ...     alpha_factors = [torch.tensor([2/3,1,3/2])],
        ...     verbose = True,
        ...     verbose_times = False,
        ...     store_data_iters = None,
        ...     store_all_data = True,
        ...     )
            iter i     | losses_quantiles                                          | lams_quantiles                                            | alphas_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 5.5e+00   | 2.5e+01   | 5.3e+01   | 1.3e+02   | 5.1e+02   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 7.4e-02   | 6.4e-01   | 1.1e+00   | 2.2e+00   | 7.7e+01   | 2.5e-07   | 2.5e-07   | 2.5e-07   | 2.5e-07   | 4.0e-06   | 6.7e-01   | 6.7e-01   | 1.0e+00   | 1.1e+00   | 1.5e+00   
            2          | 1.0e-05   | 6.9e-04   | 1.8e-03   | 3.9e-03   | 1.2e+00   | 6.2e-08   | 6.2e-08   | 1.0e-06   | 1.0e-06   | 1.0e-06   | 6.7e-01   | 6.7e-01   | 1.0e+00   | 1.1e+00   | 1.5e+00   
        >>> torch.allclose(theta_hat,theta_true,atol=5e-2)
        False
        >>> print_data_signatures(data)
            data['theta'].shape = (4, 4, 2, 2)
            data['iterrange'].shape = (3,)
            data['times'].shape = (3,)
            data['losses_quantiles']
                data['losses_quantiles']['0'].shape = (3,)
                data['losses_quantiles']['1'].shape = (3,)
                data['losses_quantiles']['5'].shape = (3,)
                data['losses_quantiles']['10'].shape = (3,)
                data['losses_quantiles']['25'].shape = (3,)
                data['losses_quantiles']['40'].shape = (3,)
                data['losses_quantiles']['50'].shape = (3,)
                data['losses_quantiles']['60'].shape = (3,)
                data['losses_quantiles']['75'].shape = (3,)
                data['losses_quantiles']['90'].shape = (3,)
                data['losses_quantiles']['95'].shape = (3,)
                data['losses_quantiles']['99'].shape = (3,)
                data['losses_quantiles']['100'].shape = (3,)
            data['lams_quantiles']
                data['lams_quantiles']['0'].shape = (3,)
                data['lams_quantiles']['1'].shape = (3,)
                data['lams_quantiles']['5'].shape = (3,)
                data['lams_quantiles']['10'].shape = (3,)
                data['lams_quantiles']['25'].shape = (3,)
                data['lams_quantiles']['40'].shape = (3,)
                data['lams_quantiles']['50'].shape = (3,)
                data['lams_quantiles']['60'].shape = (3,)
                data['lams_quantiles']['75'].shape = (3,)
                data['lams_quantiles']['90'].shape = (3,)
                data['lams_quantiles']['95'].shape = (3,)
                data['lams_quantiles']['99'].shape = (3,)
                data['lams_quantiles']['100'].shape = (3,)
            data['alphas_quantiles']
                data['alphas_quantiles']['0'].shape = (3,)
                data['alphas_quantiles']['1'].shape = (3,)
                data['alphas_quantiles']['5'].shape = (3,)
                data['alphas_quantiles']['10'].shape = (3,)
                data['alphas_quantiles']['25'].shape = (3,)
                data['alphas_quantiles']['40'].shape = (3,)
                data['alphas_quantiles']['50'].shape = (3,)
                data['alphas_quantiles']['60'].shape = (3,)
                data['alphas_quantiles']['75'].shape = (3,)
                data['alphas_quantiles']['90'].shape = (3,)
                data['alphas_quantiles']['95'].shape = (3,)
                data['alphas_quantiles']['99'].shape = (3,)
                data['alphas_quantiles']['100'].shape = (3,)
            data['thetas'].shape = (3, 4, 4, 2, 2)
            data['losses'].shape = (3, 4, 4)
            data['lams'].shape = (3, 4, 4)
            data['alphas'].shape = (3, 4, 4)
    """
    if warn and (not torch.get_default_dtype()==torch.float64): warnings.warn('''
            torch.get_default_dtype() = %s, but lm_opt often requires high precision updates. We recommend using:
                torch.set_default_dtype(torch.float64)'''%str(torch.get_default_dtype()))
    assert torch.get_default_dtype() in [torch.float32,torch.float64]
    default_dtype = torch.get_default_dtype()
    device = str(theta0.device)
    default_device = str(torch.get_default_device())
    assert iters%1==0, "iters should be an int"
    assert iters>=0
    assert callable(f) 
    assert batch_dims>=0
    assert isinstance(theta0,torch.Tensor)
    batch_shape = tuple(theta0.shape[:batch_dims])
    R = int(torch.tensor(batch_shape).prod())
    nonbatch_theta_dims = theta0.ndim-batch_dims
    nonbatch_theta_shape = tuple(theta0.shape[batch_dims:])
    nonbatch_y_dims = ytrue.ndim-batch_dims
    nonbatch_y_shape = tuple(ytrue.shape[batch_dims:])
    K = int(torch.tensor(nonbatch_y_shape).prod())
    T = int(torch.tensor(nonbatch_theta_shape).prod())
    if batch_dims==0:
        theta = theta0[None,...]
    else: # batch_dims>0:
        theta = theta0.flatten(end_dim=batch_dims-1)
    if batch_dims==0:
        ytrue = ytrue[None,...]
    else: # batch_dims>0:
        ytrue = ytrue.flatten(end_dim=batch_dims-1)
    assert isinstance(f_kwargs_vec,dict)
    assert isinstance(f_kwargs_no_vec,dict)
    f_kwargs_vec_names = list(f_kwargs_vec.keys())
    f_kwargs_vec_vals = []
    for key in f_kwargs_vec_names:
        assert f_kwargs_vec[key].shape[:batch_dims]==batch_shape, "f_kwargs_vec['%s'].shape[:%d] = %s but bs = %s"%(key,batch_dims,f_kwargs_vec[key].shape[:batch_dims])
        if batch_dims==0:
            f_kwargs_vec_vals.append(f_kwargs_vec[key][None,...])
        else: # batch_dims>0
            f_kwargs_vec_vals.append(f_kwargs_vec[key].flatten(end_dim=batch_dims-1))
    f_kwargs_vec_names = ["ytrue"]+f_kwargs_vec_names
    f_kwargs_vec_vals = [ytrue]+f_kwargs_vec_vals
    if verbose is None: 
        verbose = max(1,iters//20)
    assert verbose%1==0
    assert verbose>=0 
    if store_data_iters is None: 
        store_data_iters = max(1,iters//1000)
    assert store_data_iters%1==0
    assert store_data_iters>=0 
    assert isinstance(store_all_data,bool)
    loss_mult = float(loss_mult)
    assert loss_mult!=0 
    signminimize = -1 if loss_mult<0 else 1
    loss_shift = float(loss_shift)
    if residtol is None: 
        if default_dtype==torch.float64:
            residtol = 1e-12
        elif default_dtype==torch.float32:
            residtol = 2.5e-4
        else:
            raise Exception("default_dtype = %s not parsed"%str(default_dtype))
    assert residtol>=0
    assert lam0>0
    assert alpha0>0
    if np.isscalar(lam_factors):
        lam_factors = [torch.tensor([lam_factors],device=device)]
    elif isinstance(lam_factors,torch.Tensor):
        lam_factors = [lam_factors.to(device)]
    lam_factors = [torch.tensor(list(lam_factors[i])).to(device) for i in range(len(lam_factors))]
    assert isinstance(lam_factors,list)
    assert all(isinstance(lam_factors[i],torch.Tensor) for i in range(len(lam_factors)))
    assert all(lam_factors[i].ndim==1 for i in range(len(lam_factors)))
    if np.isscalar(alpha_factors):
        alpha_factors = [torch.tensor([alpha_factors],device=device)]
    elif isinstance(alpha_factors,torch.Tensor):
        alpha_factors = [alpha_factors.to(device)]
    alpha_factors = [torch.tensor(list(alpha_factors[i])).to(device) for i in range(len(alpha_factors))]
    assert isinstance(alpha_factors,list)
    assert all(isinstance(alpha_factors[i],torch.Tensor) for i in range(len(alpha_factors)))
    assert all(alpha_factors[i].ndim==1 for i in range(len(alpha_factors)))
    if len(alpha_factors)==1:
        alpha_factors = alpha_factors*len(lam_factors)
    if len(lam_factors)==1:
        lam_factors = lam_factors*len(alpha_factors)
    assert len(lam_factors)==len(alpha_factors)
    assert isinstance(quantiles_losses,list)
    assert all(0<=qt<=100 for qt in quantiles_losses)
    assert isinstance(quantiles_lams,list)
    assert all(0<=qt<=100 for qt in quantiles_lams)
    assert isinstance(quantiles_alphas,list)
    assert all(0<=qt<=100 for qt in quantiles_alphas)
    assert isinstance(verbose_quantiles_losses,list)
    assert all(qt in quantiles_losses for qt in verbose_quantiles_losses)
    assert isinstance(verbose_quantiles_lams,list)
    assert all(qt in quantiles_lams for qt in verbose_quantiles_lams)
    assert isinstance(verbose_quantiles_alphas,list)
    assert all(qt in quantiles_alphas for qt in verbose_quantiles_alphas)
    assert verbose_indent%1==0 
    assert verbose_indent>=0
    assert isinstance(verbose_times,bool)
    if store_data_iters:
        iterrange = []
        times = []
        losses = []
        losses_quantiles = {str(qt):[] for qt in quantiles_losses}
        lams_quantiles = {str(qt):[] for qt in quantiles_lams}
        alphas_quantiles = {str(qt):[] for qt in quantiles_alphas}
        if store_all_data:
            thetas = []
            lams = []
            alphas = []
    def f_resid(theta, *f_kwargs_vec_vals):
        assert len(f_kwargs_vec_vals)==len(f_kwargs_vec_names)
        ytrue = f_kwargs_vec_vals[0]
        f_kwargs_vec = {f_kwargs_vec_names[i]:f_kwargs_vec_vals[i] for i in range(1,len(f_kwargs_vec_names))}
        all_args = f(theta,**f_kwargs_vec,**f_kwargs_no_vec)
        assert isinstance(all_args,tuple) or isinstance(all_args,torch.Tensor), "f must return a tuple or torch.Tensor"
        if isinstance(all_args,torch.Tensor):
            yhat = all_args
            args = ()
        else: #  isinstance(all_args,tuple)
            assert all(isinstance(arg,torch.Tensor) for arg in all_args)
            yhat = all_args[0]
            args = all_args[1:]
        resid = yhat-ytrue
        return resid,(resid,yhat,*args)
    assert jacmode in ["auto","fwd","rev"]
    if jacmode=="auto":
        if T<=K:
            jacfunc = torch.func.jacfwd
        else:
            jacfunc = torch.func.jacrev
    elif jacmode=="fwd":
        jacfunc = torch.func.jacfwd
        if warn and T>K: warnings.warn('''
            For T the number of inputs and K the number of outputs:
                torch.func.jacfwd performs best when T << K. 
                torch.func.jacrev performs best when K << T.
            You are using jacmode=='fwd' but T = %d > %d = K. 
            Try using jacmode=='rev' by setting jacfwd = False.'''%(T,K))
    elif jacmode=="rev":
        jacfunc = torch.func.jacrev
        if warn and T<K: warnings.warn('''
            For T the number of inputs and K the number of outputs:
                torch.func.jacfwd performs best when T << K. 
                torch.func.jacrev performs best when K << T.
            You are using jacmode=='rev' but T = %d < %d = K. 
            Try using jacmode=='fwd' for better performance.'''%(T,K))
    jac_ftilde = jacfunc(f_resid,argnums=(0,),has_aux=True)
    vjac_ftilde = torch.func.vmap(jac_ftilde,in_dims=(0,)+(0,)*len(f_kwargs_vec_names),chunk_size=vmap_chunk_size)
    eyeT = torch.eye(T,device=device)
    Rrange = torch.arange(R,device=device)
    lam = lam0*torch.ones(R,device=device)
    alpha = alpha0*torch.ones(R,device=device)
    if verbose:
        _h_iter = "%-10s "%"iter i"
        _h_times = "| %-10s"%"times" if verbose_times else ""
        _s_losses_qt = ("| %-9s "*len(verbose_quantiles_losses))%tuple(str(qt) for qt in verbose_quantiles_losses)
        _s_lams_qt = ("| %-9s "*len(verbose_quantiles_lams))%tuple(str(qt) for qt in verbose_quantiles_lams)
        _s_alphas_qt = ("| %-9s "*len(verbose_quantiles_alphas))%tuple(str(qt) for qt in verbose_quantiles_alphas)
        _h_losses_qt = "| losses_quantiles"+" "*(len(_s_losses_qt)-len("| losses_quantiles"))
        _h_lams_qt   = "| lams_quantiles"  +" "*(len(_s_lams_qt)  -len("| lams_quantiles"))
        _h_alphas_qt = "| alphas_quantiles"+" "*(len(_s_alphas_qt)-len("| alphas_quantiles"))
        _h = _h_iter+_h_losses_qt+_h_lams_qt+_h_alphas_qt+_h_times
        _s = " "*len(_h_iter)+_s_losses_qt+_s_lams_qt+_s_alphas_qt+("|"+" "*(len(_h_times)-1) if verbose_times else " "*len(_h_times))
        print(" "*verbose_indent+_h)
        print(" "*verbose_indent+_s)
        print(" "*verbose_indent+"~"*len(_s))
    timer = Timer(device=device)
    timer.tic()
    for i in range(iters+1):
        if i==iters:
            _,(resid,yhat,*args) = f_resid(theta,*f_kwargs_vec_vals)
        else:
            (Jfull,),(resid,yhat,*args) = vjac_ftilde(theta,*f_kwargs_vec_vals)
        assert Jfull.shape==(R,*nonbatch_y_shape,*nonbatch_theta_shape)
        breakcond = i==iters or resid.abs().amax()<=residtol
        loss = loss_mult*(resid**2).flatten(start_dim=1).sum(-1)+loss_shift
        times_i = timer.toc()
        losses_quantiles_i = {str(qt): loss.nanquantile(qt/100) for qt in quantiles_losses}
        lams_quantiles_i = {str(qt): lam.nanquantile(qt/100) for qt in quantiles_lams}
        alphas_quantiles_i = {str(qt): alpha.nanquantile(qt/100) for qt in quantiles_alphas}
        if store_data_iters and (i%store_data_iters==0 or breakcond):
            iterrange.append(i)
            losses.append(loss.reshape(batch_shape).to(default_device))
            times.append(times_i)
            for qt in quantiles_losses:
                losses_quantiles[str(qt)].append(losses_quantiles_i[str(qt)].to(default_device))
            for qt in quantiles_lams:
                lams_quantiles[str(qt)].append(lams_quantiles_i[str(qt)].to(default_device))
            for qt in quantiles_alphas:
                alphas_quantiles[str(qt)].append(alphas_quantiles_i[str(qt)].to(default_device))
            if store_all_data:
                thetas.append(theta.reshape(*batch_shape,*nonbatch_theta_shape).to(default_device))
                lams.append(lam.reshape(batch_shape).to(default_device))
                alphas.append(alpha.reshape(batch_shape).to(default_device))
        if verbose and (i%verbose==0 or i==iters):
            _s_iter = "%-10d "%i
            _s_losses_qt = ("| %-9.1e "*len(verbose_quantiles_losses))%tuple(losses_quantiles_i[str(qt)] for qt in verbose_quantiles_losses)
            _s_lams_qt = ("| %-9.1e "*len(verbose_quantiles_lams))%tuple(lams_quantiles_i[str(qt)] for qt in verbose_quantiles_lams)
            _s_alphas_qt = ("| %-9.1e "*len(verbose_quantiles_alphas))%tuple(alphas_quantiles_i[str(qt)] for qt in verbose_quantiles_alphas)
            _s_times = "| %-10.1f "%(times_i) if verbose_times else ""
            print(" "*verbose_indent+_s_iter+_s_losses_qt+_s_lams_qt+_s_alphas_qt+_s_times)
        if breakcond: break
        J = Jfull.reshape((R,K,T))
        residf = resid.reshape((R,K))
        gamma = torch.einsum("rij,ri->rj",J,residf)
        JtJ = torch.einsum("rij,ril->rjl",J,J) # (R,T,T)
        lam_factors_i = lam_factors[i%len(lam_factors)]
        alpha_factors_i = alpha_factors[i%len(alpha_factors)]
        Q_lams = len(lam_factors_i)
        Q_alphas = len(alpha_factors_i)
        lams_try = lam_factors_i[:,None]*lam # (Q_lams,R)
        alphas_try = alpha_factors_i[:,None]*alpha # (Q_alphas,R)
        JtJplam = JtJ[None,:,:,:]+lams_try[:,:,None,None]*eyeT # (Q_lams,R,T,T)
        L,fails = torch.linalg.cholesky_ex(JtJplam,upper=False) # L.shape==(Q_lams,R,T,T) and fails.shape==(Q_lams,R)
        success = ~fails.to(bool) # (Q_lams,R)
        deltaf = torch.nan*torch.ones((Q_lams,R,T),device=device)
        gammas = torch.ones((Q_lams,1,1),device=device)*gamma[None,:,:] # (Q_lams,R,T)
        deltaf[success] = torch.linalg.solve_triangular(L[success].transpose(dim0=-2,dim1=-1),torch.linalg.solve_triangular(L[success],gammas[success].unsqueeze(-1),upper=False),upper=True)[...,0] # (Q_lam,R,T)
        thetasf = torch.ones((Q_lams,1,1),device=device)*theta.reshape((1,R,T)) # (Q_lam,R,T)
        thetasf_new = torch.nan*torch.ones((Q_alphas,Q_lams,R,T),device=device)
        thetasf_new[:,success] = thetasf[success]-signminimize*alpha_factors_i[:,None,None]*deltaf[success]
        thetas_new = thetasf_new.reshape((Q_alphas,Q_lams,R,*nonbatch_theta_shape))
        f_kwargs_vec_vals_success = [(torch.ones((Q_alphas,Q_lams)+(1,)*f_kwargs_vec_vals[l].ndim,device=device)*f_kwargs_vec_vals[l][None,None,...])[:,success] for l in range(len(f_kwargs_vec_vals))]
        residf_new = torch.inf*torch.ones((Q_alphas,Q_lams,R,K),device=device)
        _,(resid_new_success,*_) = f_resid(thetas_new[:,success],*f_kwargs_vec_vals_success)
        residf_new[:,success] = resid_new_success.reshape((Q_alphas,resid_new_success.size(1),K))
        losses_new = loss_mult*(residf_new**2).sum(-1)+loss_shift
        ibest = losses_new.reshape((Q_alphas*Q_lams,R)).argmin(0) # (R,)
        ibest_alpha,ibest_lam = ibest//Q_lams,ibest%Q_lams
        lam_best_new = lams_try[ibest_lam,Rrange] # (R,)
        alpha_best_new = alphas_try[ibest_alpha,Rrange] # (R,)
        loss_best_new = losses_new[ibest_alpha,ibest_lam,Rrange] # (R,)
        thetas_best_new = thetas_new[ibest_alpha,ibest_lam,Rrange] # (R,*nonbatch_theta_shape)
        improved = loss_best_new<loss # (R,)
        lam[improved] = lam_best_new[improved]
        alpha[improved] = alpha_best_new[improved]
        theta[improved] = thetas_best_new[improved]
        lam[~improved] = lams_try[:,~improved].amax(0)
        alpha[~improved] = alphas_try[:,~improved].amin(0)
    theta = theta.reshape((*batch_shape,*nonbatch_theta_shape))
    if batch_shape==():
        args = [arg[0] for arg in args]
    if store_data_iters==0:
        if args==[]:
            return theta
        else:
            return theta,*args
    else:
        data = {
            "theta": theta.to(default_device), 
            "iterrange": torch.tensor(iterrange,dtype=int), 
            "times": torch.tensor(times), 
            "losses_quantiles": {str(qt):torch.tensor(losses_quantiles[str(qt)]) for qt in quantiles_losses},
            "lams_quantiles": {str(qt):torch.tensor(lams_quantiles[str(qt)]) for qt in quantiles_lams},
            "alphas_quantiles": {str(qt):torch.tensor(alphas_quantiles[str(qt)]) for qt in quantiles_alphas},
            }
        if store_all_data:
            data["thetas"] = torch.stack(thetas,dim=0)
            data["losses"] = torch.stack(losses,dim=0)
            data["lams"] = torch.stack(lams,dim=0)
            data["alphas"] = torch.stack(alphas,dim=0)
        if args==[]:
            return theta,data
        else:
            return theta,*args,data

minres 

minres(A, B, X0=None, iters=None, residtol=None, verbose=False, verbose_indent=4, quantiles_losses=[0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100], verbose_quantiles_losses=[5, 25, 50, 75, 90], verbose_times=True, warn=True, store_data_iters=False, store_all_data=False)

MINRES algorithm for solving linear systems \(AX=B\) where \(A\) is real-symmetric or complex-Hermitian

A translation of scipy.sparse.linalg.minres.

Parameters:

Name Type Description Default
A Union[Tensor, callable]

Symmetric matrix A with shape (...,n,n), or
callable(A) where a(x) should return the batch matrix multiplication of A and X,

required
B Tensor

Right hand side tensor \(B\) with shape (...,n,k)

 required
X0 Tensor

Initial guess for \(X\) with shape (...,n,k), defaults to zeros.

None
iters int

number of minres iterations, defaults to 5n.

None
residtol float

Non-negative tolerance on the maximum residual for early stopping, defaults to 1e-12 for torch.float64 and 2.5e-4 for torch.float32.

 None
verbose int

Controls logging verbosity

  • If True, perform logging.
  • If a positive int, only log every verbose iterations.
  • If None, set to a reasonable positive int based on the maximum number of iterations
  • If False, don't log.
False
verbose_indent int

Non-negative number of indentation spaces for logging.

 4
quantiles_losses list

Loss quantiles to record.

 [0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100]
verbose_quantiles_losses list

Loss quantiles to show in verbose log.

 [5, 25, 50, 75, 90]
verbose_times bool

If False, do not show the times in the verbose log. This is mostly for testing where timing is not reproducible.

True
warn bool

If False, then suppress warnings.

 True
store_data_iters int

Controls storage iterations with the same options as verbose. If store_data_iters==0, then the data is not collected or returned.

  • If True, store every iteration.
  • If a positive int, only store every store_data_iters iterations.
  • If None, set to a reasonable positive int based on the maximum number of iterations
  • If False, don't store data, and do not return data
False
store_all_data bool

If True, store the x values as well as the metrics.

False

Returns:

Name Type Description
x Tensor

Optimized \(X\).
data dict

Iteration data, only returned when store_data_iters>0

Examples:

>>> torch.set_default_dtype(torch.float64)
>>> rng = torch.Generator().manual_seed(7)

Real-symmetric example with column vector \(b\) 

>>> n = 5
>>> A = torch.randn(n,n,generator=rng)
>>> A = (A+A.T)/2
>>> b = torch.rand(n,generator=rng)
>>> x_true = torch.linalg.solve(A,b[...,None])[...,0]
>>> x_true
tensor([-0.1402,  0.4565,  0.2920,  0.2470,  0.3251])
>>> torch.allclose(A@x_true-b,torch.zeros_like(b))
True
>>> x_minres = minres(A,b[...,None],verbose=None,verbose_times=False)[...,0]
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 6.1e-01   | 6.1e-01   | 6.1e-01   | 6.1e-01   | 6.1e-01   
    2          | 2.1e-01   | 2.1e-01   | 2.1e-01   | 2.1e-01   | 2.1e-01   
    3          | 1.9e-01   | 1.9e-01   | 1.9e-01   | 1.9e-01   | 1.9e-01   
    4          | 7.3e-02   | 7.3e-02   | 7.3e-02   | 7.3e-02   | 7.3e-02   
    5          | 5.4e-16   | 5.4e-16   | 5.4e-16   | 5.4e-16   | 5.4e-16   
>>> torch.allclose(x_minres,x_true)
True

Complex-Hermitian example with column vector \(b\) 

>>> n = 5
>>> A = torch.randn(n,n,dtype=torch.complex128,generator=rng)
>>> A = (A+A.adjoint())/2
>>> b = torch.rand(n,dtype=torch.complex128,generator=rng)
>>> x_true = torch.linalg.solve(A,b[...,None])[...,0]
>>> x_true
tensor([ 0.2207+0.2879j,  0.0928-0.0057j,  0.2681+1.3488j, -1.2520-0.4214j,
        -0.8860-0.6922j])
>>> torch.allclose(A@x_true-b,torch.zeros_like(b))
True
>>> x_minres = minres(A,b[...,None],verbose=None,verbose_times=False)[...,0]
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 8.7e-01   | 8.7e-01   | 8.7e-01   | 8.7e-01   | 8.7e-01   
    2          | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   
    3          | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   
    4          | 2.6e-01   | 2.6e-01   | 2.6e-01   | 2.6e-01   | 2.6e-01   
    5          | 3.0e-15   | 3.0e-15   | 3.0e-15   | 3.0e-15   | 3.0e-15   
>>> torch.allclose(x_minres,x_true)
True

Matrix \(B\)

>>> n = 5
>>> k = 3
>>> A = torch.randn(n,n,generator=rng)
>>> A = (A+A.T)/2
>>> B = torch.rand(n,k,generator=rng)
>>> X_true = torch.linalg.solve(A,B)
>>> X_true
tensor([[ 0.8801, -0.0116,  0.4805],
        [-1.1095, -1.6166, -0.7103],
        [-2.9918, -1.9201, -3.5855],
        [-4.1777, -3.6586, -5.1658],
        [ 1.5417,  0.9814,  1.3790]])
>>> torch.allclose(A@X_true-B,torch.zeros_like(B))
True
>>> X_minres = minres(A,B,verbose=None,verbose_times=False)
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 6.6e-01   | 7.5e-01   | 8.7e-01   | 8.8e-01   | 8.9e-01   
    2          | 6.2e-01   | 6.4e-01   | 6.6e-01   | 7.3e-01   | 7.6e-01   
    3          | 3.5e-01   | 4.7e-01   | 6.1e-01   | 6.3e-01   | 6.4e-01   
    4          | 1.5e-01   | 1.8e-01   | 2.2e-01   | 3.4e-01   | 4.1e-01   
    5          | 7.0e-15   | 9.1e-15   | 1.2e-14   | 2.2e-14   | 2.8e-14   
>>> torch.allclose(X_minres,X_true)
True

Tri-diagonal \(A\) with storage-saving multiplication function 

>>> n = 5
>>> k = 3
>>> A_diag = torch.randn(n,generator=rng)
>>> A_off_diag = torch.randn(n-1,generator=rng) 
>>> A = torch.zeros(n,n)
>>> A[torch.arange(n),torch.arange(n)] = A_diag 
>>> A[torch.arange(n-1),torch.arange(1,n)] = A_off_diag
>>> A[torch.arange(1,n),torch.arange(n-1)] = A_off_diag
>>> A
tensor([[-0.2728, -0.1545,  0.0000,  0.0000,  0.0000],
        [-0.1545, -0.0275, -0.0120,  0.0000,  0.0000],
        [ 0.0000, -0.0120, -0.4436,  0.2802,  0.0000],
        [ 0.0000,  0.0000,  0.2802, -0.7303,  0.9724],
        [ 0.0000,  0.0000,  0.0000,  0.9724, -0.4180]])
>>> B = torch.rand(n,k,generator=rng)
>>> X_true = torch.linalg.solve(A,B)
>>> X_true
tensor([[-5.5996, -3.1344, -6.0989],
        [ 6.5450, -0.5211,  5.0505],
        [-1.8303, -0.9818, -0.4647],
        [ 0.5862,  0.9233,  1.2342],
        [ 1.3614,  1.4523,  1.9166]])
>>> torch.allclose(A@X_true-B,torch.zeros_like(B))
True
>>> def A_mult(x):
...     y = x*A_diag[:,None]
...     y[1:,:] += x[:-1,:]*A_off_diag[:,None]
...     y[:-1,:] += x[1:,:]*A_off_diag[:,None]
...     return y
>>> torch.allclose(A_mult(X_true),A@X_true)
True
>>> X_minres = minres(A_mult,B,verbose=None,verbose_times=False)
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 7.5e-01   | 8.1e-01   | 8.8e-01   | 9.3e-01   | 9.6e-01   
    2          | 4.5e-01   | 5.6e-01   | 7.1e-01   | 8.0e-01   | 8.5e-01   
    3          | 1.5e-01   | 1.9e-01   | 2.4e-01   | 2.9e-01   | 3.2e-01   
    4          | 5.7e-02   | 1.4e-01   | 2.4e-01   | 2.9e-01   | 3.2e-01   
    5          | 4.0e-15   | 5.4e-15   | 7.1e-15   | 1.9e-14   | 2.6e-14   
>>> torch.allclose(X_minres,X_true)
True

Batched tri-diagonal \(A\) with storage-saving multiplication function 

>>> n = 100
>>> k = 3
>>> A_diag = torch.randn(2,1,4,n,generator=rng)
>>> A_off_diag = torch.randn(2,1,4,n-1,generator=rng) 
>>> A = torch.zeros(2,1,4,n,n)
>>> A[...,torch.arange(n),torch.arange(n)] = A_diag 
>>> A[...,torch.arange(n-1),torch.arange(1,n)] = A_off_diag
>>> A[...,torch.arange(1,n),torch.arange(n-1)] = A_off_diag
>>> B = torch.rand(2,6,1,n,k,generator=rng)
>>> X_true = torch.linalg.solve(A,B)
>>> torch.allclose(torch.einsum("...ij,...jk->...ik",A,X_true)-B,torch.zeros_like(B))
True
>>> def A_mult(x):
...     y = x*A_diag[...,:,None]
...     y[...,1:,:] += x[...,:-1,:]*A_off_diag[...,:,None]
...     y[...,:-1,:] += x[...,1:,:]*A_off_diag[...,:,None]
...     return y
>>> torch.allclose(A_mult(X_true),torch.einsum("...ij,...jk->...ik",A,X_true))
True
>>> X_minres,data = minres(A_mult,B,verbose=None,verbose_times=False,store_data_iters=None,store_all_data=True)
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    25         | 1.8e-01   | 2.3e-01   | 2.6e-01   | 2.9e-01   | 3.2e-01   
    50         | 9.4e-02   | 1.2e-01   | 1.5e-01   | 1.9e-01   | 2.1e-01   
    75         | 3.3e-02   | 5.1e-02   | 8.2e-02   | 1.5e-01   | 1.8e-01   
    100        | 7.8e-03   | 1.4e-02   | 3.8e-02   | 1.2e-01   | 1.5e-01   
    125        | 9.5e-07   | 1.2e-04   | 5.6e-04   | 2.3e-02   | 5.1e-02   
    150        | 1.2e-14   | 2.6e-14   | 1.7e-12   | 2.5e-10   | 3.1e-09   
    175        | 2.2e-15   | 3.7e-15   | 1.4e-14   | 6.5e-14   | 1.6e-13   
    176        | 2.2e-15   | 3.7e-15   | 1.3e-14   | 6.0e-14   | 1.4e-13   
>>> X_minres.shape
torch.Size([2, 6, 4, 100, 3])
>>> torch.allclose(X_minres,X_true)
True
>>> print_data_signatures(data)
    data['x'].shape = (2, 6, 4, 100, 3)
    data['iterrange'].shape = (177,)
    data['times'].shape = (177,)
    data['losses_quantiles']
        data['losses_quantiles']['0'].shape = (177,)
        data['losses_quantiles']['1'].shape = (177,)
        data['losses_quantiles']['5'].shape = (177,)
        data['losses_quantiles']['10'].shape = (177,)
        data['losses_quantiles']['25'].shape = (177,)
        data['losses_quantiles']['40'].shape = (177,)
        data['losses_quantiles']['50'].shape = (177,)
        data['losses_quantiles']['60'].shape = (177,)
        data['losses_quantiles']['75'].shape = (177,)
        data['losses_quantiles']['90'].shape = (177,)
        data['losses_quantiles']['95'].shape = (177,)
        data['losses_quantiles']['99'].shape = (177,)
        data['losses_quantiles']['100'].shape = (177,)
    data['xs'].shape = (177, 2, 6, 4, 100, 3)
    data['losses'].shape = (177, 2, 6, 4, 3)
Source code in agsutil/algos.py 
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def minres(
        A,
        B,
        X0 = None,
        iters = None,
        residtol = None,
        verbose = False, 
        verbose_indent = 4,
        quantiles_losses = [0,1,5,10,25,40,50,60,75,90,95,99,100],
        verbose_quantiles_losses = [5,25,50,75,90],
        verbose_times = True, 
        warn = True,
        store_data_iters = False, 
        store_all_data = False,
        ):
    r"""
    [MINRES algorithm](https://en.wikipedia.org/wiki/Minimal_residual_method) for solving linear systems $AX=B$ where $A$ is real-symmetric or complex-Hermitian

    A translation of [`scipy.sparse.linalg.minres`](https://github.com/scipy/scipy/blob/v1.17.0/scipy/sparse/linalg/_isolve/minres.py).

    Args:
        A (Union[torch.Tensor,callable]): Symmetric matrix `A` with shape `(...,n,n)`, or  
            `callable(A)` where `a(x)` should return the batch matrix multiplication of `A` and `X`,  
        B (torch.Tensor): Right hand side tensor $B$ with shape `(...,n,k)`
        X0 (torch.Tensor): Initial guess for $X$ with shape `(...,n,k)`, defaults to zeros. 
        iters (int): number of minres iterations, defaults to `5n`. 
        residtol (float): Non-negative tolerance on the maximum residual for early stopping, defaults to `1e-12` for `torch.float64` and `2.5e-4` for `torch.float32`.
        verbose (int): Controls logging verbosity

            - If `True`, perform logging. 
            - If a positive int, only log every `verbose` iterations. 
            - If `None`, set to a reasonable positive int based on the maximum number of iterations
            - If `False`, don't log. 

        verbose_indent (int): Non-negative number of indentation spaces for logging.
        quantiles_losses (list): Loss quantiles to record.
        verbose_quantiles_losses (list): Loss quantiles to show in verbose log.
        verbose_times (bool): If `False`, do not show the times in the verbose log. This is mostly for testing where timing is not reproducible. 
        warn (bool): If `False`, then suppress warnings.
        store_data_iters (int): Controls storage iterations with the same options as verbose. If `store_data_iters==0`, then the data is not collected or returned. 

            - If `True`, store every iteration. 
            - If a positive int, only store every `store_data_iters` iterations. 
            - If `None`, set to a reasonable positive int based on the maximum number of iterations
            - If `False`, don't store data, and do not return data 

        store_all_data (bool): If `True`, store the `x` values as well as the metrics. 

    Returns:
        x (torch.Tensor): Optimized $X$.
        data (dict): Iteration data, only returned when `store_data_iters>0`

    Examples:

        >>> torch.set_default_dtype(torch.float64)
        >>> rng = torch.Generator().manual_seed(7)

    Real-symmetric example with column vector $b$ 

        >>> n = 5
        >>> A = torch.randn(n,n,generator=rng)
        >>> A = (A+A.T)/2
        >>> b = torch.rand(n,generator=rng)
        >>> x_true = torch.linalg.solve(A,b[...,None])[...,0]
        >>> x_true
        tensor([-0.1402,  0.4565,  0.2920,  0.2470,  0.3251])
        >>> torch.allclose(A@x_true-b,torch.zeros_like(b))
        True
        >>> x_minres = minres(A,b[...,None],verbose=None,verbose_times=False)[...,0]
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 6.1e-01   | 6.1e-01   | 6.1e-01   | 6.1e-01   | 6.1e-01   
            2          | 2.1e-01   | 2.1e-01   | 2.1e-01   | 2.1e-01   | 2.1e-01   
            3          | 1.9e-01   | 1.9e-01   | 1.9e-01   | 1.9e-01   | 1.9e-01   
            4          | 7.3e-02   | 7.3e-02   | 7.3e-02   | 7.3e-02   | 7.3e-02   
            5          | 5.4e-16   | 5.4e-16   | 5.4e-16   | 5.4e-16   | 5.4e-16   
        >>> torch.allclose(x_minres,x_true)
        True

    Complex-Hermitian example with column vector $b$ 

        >>> n = 5
        >>> A = torch.randn(n,n,dtype=torch.complex128,generator=rng)
        >>> A = (A+A.adjoint())/2
        >>> b = torch.rand(n,dtype=torch.complex128,generator=rng)
        >>> x_true = torch.linalg.solve(A,b[...,None])[...,0]
        >>> x_true
        tensor([ 0.2207+0.2879j,  0.0928-0.0057j,  0.2681+1.3488j, -1.2520-0.4214j,
                -0.8860-0.6922j])
        >>> torch.allclose(A@x_true-b,torch.zeros_like(b))
        True
        >>> x_minres = minres(A,b[...,None],verbose=None,verbose_times=False)[...,0]
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 8.7e-01   | 8.7e-01   | 8.7e-01   | 8.7e-01   | 8.7e-01   
            2          | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   
            3          | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   | 5.4e-01   
            4          | 2.6e-01   | 2.6e-01   | 2.6e-01   | 2.6e-01   | 2.6e-01   
            5          | 3.0e-15   | 3.0e-15   | 3.0e-15   | 3.0e-15   | 3.0e-15   
        >>> torch.allclose(x_minres,x_true)
        True

    Matrix $B$

        >>> n = 5
        >>> k = 3
        >>> A = torch.randn(n,n,generator=rng)
        >>> A = (A+A.T)/2
        >>> B = torch.rand(n,k,generator=rng)
        >>> X_true = torch.linalg.solve(A,B)
        >>> X_true
        tensor([[ 0.8801, -0.0116,  0.4805],
                [-1.1095, -1.6166, -0.7103],
                [-2.9918, -1.9201, -3.5855],
                [-4.1777, -3.6586, -5.1658],
                [ 1.5417,  0.9814,  1.3790]])
        >>> torch.allclose(A@X_true-B,torch.zeros_like(B))
        True
        >>> X_minres = minres(A,B,verbose=None,verbose_times=False)
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 6.6e-01   | 7.5e-01   | 8.7e-01   | 8.8e-01   | 8.9e-01   
            2          | 6.2e-01   | 6.4e-01   | 6.6e-01   | 7.3e-01   | 7.6e-01   
            3          | 3.5e-01   | 4.7e-01   | 6.1e-01   | 6.3e-01   | 6.4e-01   
            4          | 1.5e-01   | 1.8e-01   | 2.2e-01   | 3.4e-01   | 4.1e-01   
            5          | 7.0e-15   | 9.1e-15   | 1.2e-14   | 2.2e-14   | 2.8e-14   
        >>> torch.allclose(X_minres,X_true)
        True

    Tri-diagonal $A$ with storage-saving multiplication function 

        >>> n = 5
        >>> k = 3
        >>> A_diag = torch.randn(n,generator=rng)
        >>> A_off_diag = torch.randn(n-1,generator=rng) 
        >>> A = torch.zeros(n,n)
        >>> A[torch.arange(n),torch.arange(n)] = A_diag 
        >>> A[torch.arange(n-1),torch.arange(1,n)] = A_off_diag
        >>> A[torch.arange(1,n),torch.arange(n-1)] = A_off_diag
        >>> A
        tensor([[-0.2728, -0.1545,  0.0000,  0.0000,  0.0000],
                [-0.1545, -0.0275, -0.0120,  0.0000,  0.0000],
                [ 0.0000, -0.0120, -0.4436,  0.2802,  0.0000],
                [ 0.0000,  0.0000,  0.2802, -0.7303,  0.9724],
                [ 0.0000,  0.0000,  0.0000,  0.9724, -0.4180]])
        >>> B = torch.rand(n,k,generator=rng)
        >>> X_true = torch.linalg.solve(A,B)
        >>> X_true
        tensor([[-5.5996, -3.1344, -6.0989],
                [ 6.5450, -0.5211,  5.0505],
                [-1.8303, -0.9818, -0.4647],
                [ 0.5862,  0.9233,  1.2342],
                [ 1.3614,  1.4523,  1.9166]])
        >>> torch.allclose(A@X_true-B,torch.zeros_like(B))
        True
        >>> def A_mult(x):
        ...     y = x*A_diag[:,None]
        ...     y[1:,:] += x[:-1,:]*A_off_diag[:,None]
        ...     y[:-1,:] += x[1:,:]*A_off_diag[:,None]
        ...     return y
        >>> torch.allclose(A_mult(X_true),A@X_true)
        True
        >>> X_minres = minres(A_mult,B,verbose=None,verbose_times=False)
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 7.5e-01   | 8.1e-01   | 8.8e-01   | 9.3e-01   | 9.6e-01   
            2          | 4.5e-01   | 5.6e-01   | 7.1e-01   | 8.0e-01   | 8.5e-01   
            3          | 1.5e-01   | 1.9e-01   | 2.4e-01   | 2.9e-01   | 3.2e-01   
            4          | 5.7e-02   | 1.4e-01   | 2.4e-01   | 2.9e-01   | 3.2e-01   
            5          | 4.0e-15   | 5.4e-15   | 7.1e-15   | 1.9e-14   | 2.6e-14   
        >>> torch.allclose(X_minres,X_true)
        True

    Batched tri-diagonal $A$ with storage-saving multiplication function 

        >>> n = 100
        >>> k = 3
        >>> A_diag = torch.randn(2,1,4,n,generator=rng)
        >>> A_off_diag = torch.randn(2,1,4,n-1,generator=rng) 
        >>> A = torch.zeros(2,1,4,n,n)
        >>> A[...,torch.arange(n),torch.arange(n)] = A_diag 
        >>> A[...,torch.arange(n-1),torch.arange(1,n)] = A_off_diag
        >>> A[...,torch.arange(1,n),torch.arange(n-1)] = A_off_diag
        >>> B = torch.rand(2,6,1,n,k,generator=rng)
        >>> X_true = torch.linalg.solve(A,B)
        >>> torch.allclose(torch.einsum("...ij,...jk->...ik",A,X_true)-B,torch.zeros_like(B))
        True
        >>> def A_mult(x):
        ...     y = x*A_diag[...,:,None]
        ...     y[...,1:,:] += x[...,:-1,:]*A_off_diag[...,:,None]
        ...     y[...,:-1,:] += x[...,1:,:]*A_off_diag[...,:,None]
        ...     return y
        >>> torch.allclose(A_mult(X_true),torch.einsum("...ij,...jk->...ik",A,X_true))
        True
        >>> X_minres,data = minres(A_mult,B,verbose=None,verbose_times=False,store_data_iters=None,store_all_data=True)
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            25         | 1.8e-01   | 2.3e-01   | 2.6e-01   | 2.9e-01   | 3.2e-01   
            50         | 9.4e-02   | 1.2e-01   | 1.5e-01   | 1.9e-01   | 2.1e-01   
            75         | 3.3e-02   | 5.1e-02   | 8.2e-02   | 1.5e-01   | 1.8e-01   
            100        | 7.8e-03   | 1.4e-02   | 3.8e-02   | 1.2e-01   | 1.5e-01   
            125        | 9.5e-07   | 1.2e-04   | 5.6e-04   | 2.3e-02   | 5.1e-02   
            150        | 1.2e-14   | 2.6e-14   | 1.7e-12   | 2.5e-10   | 3.1e-09   
            175        | 2.2e-15   | 3.7e-15   | 1.4e-14   | 6.5e-14   | 1.6e-13   
            176        | 2.2e-15   | 3.7e-15   | 1.3e-14   | 6.0e-14   | 1.4e-13   
        >>> X_minres.shape
        torch.Size([2, 6, 4, 100, 3])
        >>> torch.allclose(X_minres,X_true)
        True
        >>> print_data_signatures(data)
            data['x'].shape = (2, 6, 4, 100, 3)
            data['iterrange'].shape = (177,)
            data['times'].shape = (177,)
            data['losses_quantiles']
                data['losses_quantiles']['0'].shape = (177,)
                data['losses_quantiles']['1'].shape = (177,)
                data['losses_quantiles']['5'].shape = (177,)
                data['losses_quantiles']['10'].shape = (177,)
                data['losses_quantiles']['25'].shape = (177,)
                data['losses_quantiles']['40'].shape = (177,)
                data['losses_quantiles']['50'].shape = (177,)
                data['losses_quantiles']['60'].shape = (177,)
                data['losses_quantiles']['75'].shape = (177,)
                data['losses_quantiles']['90'].shape = (177,)
                data['losses_quantiles']['95'].shape = (177,)
                data['losses_quantiles']['99'].shape = (177,)
                data['losses_quantiles']['100'].shape = (177,)
            data['xs'].shape = (177, 2, 6, 4, 100, 3)
            data['losses'].shape = (177, 2, 6, 4, 3)
    """
    if warn and (not torch.get_default_dtype()==torch.float64): warnings.warn('''
            torch.get_default_dtype() = %s, but lm_opt often requires high precision updates. We recommend using:
                torch.set_default_dtype(torch.float64)'''%str(torch.get_default_dtype()))
    assert torch.get_default_dtype() in [torch.float32,torch.float64]
    default_dtype = torch.get_default_dtype()
    device = str(B.device)
    default_device = str(torch.get_default_device())
    assert B.ndim>=2, "B should have shape (...,n,k)"
    n = B.size(-2)
    k = B.size(-1)
    if X0 is None: 
        X0 = torch.zeros_like(B)
    if isinstance(A,torch.Tensor):
        assert A.shape[-2:]==(n,n)
        assert torch.allclose(A.adjoint(),A)
        matvec = lambda X: torch.einsum("...ij,...jk->...ik",A,X)
    else:
        assert callable(A)
        matvec = A
    if iters is None: 
        iters = 5*n 
    assert iters>=0
    assert iters%1==0
    if residtol is None: 
        if default_dtype==torch.float64:
            residtol = 1e-12
        elif default_dtype==torch.float32:
            residtol = 2.5e-4
        else:
            raise Exception("default_dtype = %s not parsed"%str(default_dtype))
    assert residtol>=0
    if verbose is None: 
        verbose = max(1,iters//20)
    assert verbose%1==0
    assert verbose>=0 
    if store_data_iters is None: 
        store_data_iters = max(1,iters//1000)
    assert store_data_iters%1==0
    assert store_data_iters>=0 
    assert isinstance(store_all_data,bool)
    assert isinstance(quantiles_losses,list)
    assert all(0<=qt<=100 for qt in quantiles_losses)
    assert isinstance(verbose_quantiles_losses,list)
    assert all(qt in quantiles_losses for qt in verbose_quantiles_losses)
    assert verbose_indent%1==0 
    assert verbose_indent>=0
    assert isinstance(verbose_times,bool)
    if verbose:
        _h_iter = "%-10s "%"iter i"
        _h_times = "| %-10s"%"times" if verbose_times else ""
        _s_losses_qt = ("| %-9s "*len(verbose_quantiles_losses))%tuple(str(qt) for qt in verbose_quantiles_losses)
        _h_losses_qt = "| losses_quantiles"+" "*(len(_s_losses_qt)-len("| losses_quantiles"))
        _h = _h_iter+_h_losses_qt+_h_times
        _s = " "*len(_h_iter)+_s_losses_qt+("|"+" "*(len(_h_times)-1) if verbose_times else " "*len(_h_times))
        print(" "*verbose_indent+_h)
        print(" "*verbose_indent+_s)
        print(" "*verbose_indent+"~"*len(_s))
    timer = Timer(device=device)
    timer.tic()
    psolve = lambda X: X # TODO: implement more involved preconditioned solver
    inner = lambda a,b: torch.einsum("...ij,...ij->...j",a.conj(),b)
    Anorm = 0
    eps = torch.finfo(B.dtype).eps
    x = X0 
    Ax = matvec(x)
    assert Ax.shape[-2:]==(n,k)
    batch_shape = tuple(Ax.shape[:-2])
    if store_data_iters:
        iterrange = []
        times = []
        losses = []
        losses_quantiles = {str(qt):[] for qt in quantiles_losses}
        if store_all_data:
            xs = []
    r1 = B-Ax # (...,n,k)
    y = psolve(r1) # (...,n,k)
    beta1 = torch.sqrt(inner(r1,y)) # (...,k)
    bnorm = torch.linalg.norm(B,dim=-2) # (...,k)
    oldb = 0
    beta = beta1
    dbar = 0
    epsln = torch.zeros(1,device=device)
    phibar = beta1
    tnorm2 = 0
    cs = -1
    sn = 0
    w = torch.zeros_like(B)
    w2 = torch.zeros_like(B)
    r2 = r1
    shift = 0 # TODO: If shift != 0 then the method solves (A - shift*I)x = b
    for i in range(iters+1):
        resid = matvec(x)-B 
        breakcond = i==iters or resid.abs().amax()<=residtol
        loss = torch.linalg.norm(resid,dim=-2)/bnorm
        times_i = timer.toc()
        losses_quantiles_i = {str(qt): loss.nanquantile(qt/100) for qt in quantiles_losses}
        if store_data_iters and (i%store_data_iters==0 or breakcond):
            iterrange.append(i)
            losses.append(loss.to(default_device))
            times.append(times_i)
            for qt in quantiles_losses:
                losses_quantiles[str(qt)].append(losses_quantiles_i[str(qt)].to(default_device))
            if store_all_data:
                xs.append(x.expand(resid.shape).to(default_device))
        if verbose and (i%verbose==0 or breakcond):
            _s_iter = "%-10d "%i
            _s_losses_qt = ("| %-9.1e "*len(verbose_quantiles_losses))%tuple(losses_quantiles_i[str(qt)] for qt in verbose_quantiles_losses)
            _s_times = "| %-10.1f "%(times_i) if verbose_times else ""
            print(" "*verbose_indent+_s_iter+_s_losses_qt+_s_times)
        if breakcond: break 
        s = 1/beta
        v = s[...,None,:]*y
        y = matvec(v)
        y = y-shift*v
        if i>0:
            y = y-(beta/oldb)[...,None,:]*r1
        alfa = inner(v,y)
        y = y-(alfa/beta)[...,None,:]*r2
        r1 = r2
        r2 = y
        y = psolve(r2)
        oldb = beta
        beta = inner(r2,y)
        beta = torch.sqrt(beta)
        tnorm2 += alfa**2+oldb**2+beta**2
        oldeps = epsln
        delta = cs*dbar+sn*alfa
        gbar = sn*dbar-cs*alfa
        epsln = sn*beta
        dbar = -cs*beta
        gamma = torch.linalg.norm(torch.stack([gbar,beta],dim=-1),dim=-1)
        gamma = torch.maximum(gamma,eps*torch.ones(1,device=device))
        cs = gbar/gamma
        sn = beta/gamma
        phi = cs*phibar
        phibar = sn*phibar
        denom = 1/gamma
        w1 = w2
        w2 = w
        w = (v-oldeps[...,None,:]*w1-delta[...,None,:]*w2)*denom[...,None,:]
        x = x+phi[...,None,:]*w
    if store_data_iters==0:
        return x 
    else:
        data = {
            "x": x.to(default_device), 
            "iterrange": torch.tensor(iterrange,dtype=int), 
            "times": torch.tensor(times), 
            "losses_quantiles": {str(qt):torch.tensor(losses_quantiles[str(qt)]) for qt in quantiles_losses},
            }
        if store_all_data:
            data["xs"] = torch.stack(xs,dim=0)
            data["losses"] = torch.stack(losses,dim=0)
        return x,data

minres_qlp_cs 

minres_qlp_cs(A, B, X0=None, iters=None, residtol=None, verbose=False, verbose_indent=4, quantiles_losses=[0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100], verbose_quantiles_losses=[5, 25, 50, 75, 90], verbose_times=True, warn=True, store_data_iters=False, store_all_data=False)

MINRES QLP algorith for complex-symmetric matrices.

A translation of the MATLAB version of MINRESQLP.

References

  1. S.-C. Choi, C. C. Paige, and M. A. Saunders, MINRES-QLP: A Krylov subspace method for indefinite or singular symmetric systems, SIAM Journal of Scientific Computing, submitted on March 7, 2010.

  2. S.-C. Choi's PhD Dissertation, Stanford University, 2006: http://www.stanford.edu/group/SOL/dissertations.html

Parameters:

Name Type Description Default
A Union[Tensor, callable]

Symmetric matrix A with shape (...,n,n), or
callable(A) where a(x) should return the batch matrix multiplication of A and X,

required
B Tensor

Right hand side tensor \(B\) with shape (...,n,k)

 required
X0 Tensor

Initial guess for \(X\) with shape (...,n,k), defaults to zeros.

None
iters int

number of minres iterations, defaults to 5n.

None
residtol float

Non-negative tolerance on the maximum residual for early stopping, defaults to 1e-12 for torch.float64 and 2.5e-4 for torch.float32.

 None
verbose int

Controls logging verbosity

  • If True, perform logging.
  • If a positive int, only log every verbose iterations.
  • If None, set to a reasonable positive int based on the maximum number of iterations
  • If False, don't log.
False
verbose_indent int

Non-negative number of indentation spaces for logging.

 4
quantiles_losses list

Loss quantiles to record.

 [0, 1, 5, 10, 25, 40, 50, 60, 75, 90, 95, 99, 100]
verbose_quantiles_losses list

Loss quantiles to show in verbose log.

 [5, 25, 50, 75, 90]
verbose_times bool

If False, do not show the times in the verbose log. This is mostly for testing where timing is not reproducible.

True
warn bool

If False, then suppress warnings.

 True
store_data_iters int

Controls storage iterations with the same options as verbose. If store_data_iters==0, then the data is not collected or returned.

  • If True, store every iteration.
  • If a positive int, only store every store_data_iters iterations.
  • If None, set to a reasonable positive int based on the maximum number of iterations
  • If False, don't store data, and do not return data
False
store_all_data bool

If True, store the x values as well as the metrics.

False

Returns:

Name Type Description
x Tensor

Optimized \(X\).
data dict

Iteration data, only returned when store_data_iters>0

Examples:

>>> torch.set_default_dtype(torch.float64)
>>> rng = torch.Generator().manual_seed(7)

Column vector \(b\) 

>>> n = 5
>>> A = torch.randn(n,n,dtype=torch.complex128,generator=rng)
>>> A = (A+A.T)/2
>>> b = torch.rand(n,dtype=torch.complex128,generator=rng)
>>> x_true = torch.linalg.solve(A,b[...,None])[...,0]
>>> x_true
tensor([-0.6207-0.4121j,  0.5221+0.3249j, -1.0952+0.8594j,  0.9080-1.2110j,
        -0.9799+0.7372j])
>>> torch.allclose(A@x_true-b,torch.zeros_like(b))
True
>>> x_minres = minres_qlp_cs(A,b[...,None],verbose=None,verbose_times=False)[...,0]
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 9.6e-01   | 9.6e-01   | 9.6e-01   | 9.6e-01   | 9.6e-01   
    2          | 5.8e-01   | 5.8e-01   | 5.8e-01   | 5.8e-01   | 5.8e-01   
    3          | 4.5e-01   | 4.5e-01   | 4.5e-01   | 4.5e-01   | 4.5e-01   
    4          | 2.8e-01   | 2.8e-01   | 2.8e-01   | 2.8e-01   | 2.8e-01   
    5          | 1.7e-14   | 1.7e-14   | 1.7e-14   | 1.7e-14   | 1.7e-14   
>>> torch.allclose(x_minres,x_true)
True

Matrix \(B\)

>>> n = 5
>>> k = 3
>>> A = torch.randn(n,n,dtype=torch.complex128,generator=rng)
>>> A = (A+A.T)/2
>>> B = torch.rand(n,k,dtype=torch.complex128,generator=rng)
>>> X_true = torch.linalg.solve(A,B)
>>> X_true
tensor([[ 0.0142+0.7190j,  0.0097+0.9734j, -0.3620+0.6413j],
        [ 0.4527+0.7455j,  0.4270+0.4941j,  0.9255+0.5685j],
        [-0.7182-1.0284j,  0.3193-0.4463j, -0.3421-0.8780j],
        [-0.5973+0.4910j, -0.8147+0.8256j, -0.8821+0.7147j],
        [-0.0926+0.5500j, -0.5192+0.2817j, -0.8496+0.6976j]])
>>> torch.allclose(A@X_true-B,torch.zeros_like(B))
True
>>> X_minres = minres_qlp_cs(A,B,verbose=None,verbose_times=False)
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 9.0e-01   | 9.1e-01   | 9.2e-01   | 9.5e-01   | 9.7e-01   
    2          | 2.2e-01   | 3.1e-01   | 4.3e-01   | 5.0e-01   | 5.5e-01   
    3          | 2.0e-01   | 2.2e-01   | 2.3e-01   | 2.9e-01   | 3.3e-01   
    4          | 3.4e-02   | 5.0e-02   | 7.1e-02   | 8.1e-02   | 8.7e-02   
    5          | 1.9e-15   | 4.2e-15   | 7.0e-15   | 1.5e-14   | 2.0e-14   
>>> torch.allclose(X_minres,X_true)
True

Tri-diagonal \(A\) with storage-saving multiplication function 

>>> n = 4
>>> k = 3
>>> A_diag = torch.randn(n,dtype=torch.complex128,generator=rng)
>>> A_off_diag = torch.randn(n-1,dtype=torch.complex128,generator=rng) 
>>> A = torch.zeros(n,n,dtype=torch.complex128)
>>> A[torch.arange(n),torch.arange(n)] = A_diag 
>>> A[torch.arange(n-1),torch.arange(1,n)] = A_off_diag
>>> A[torch.arange(1,n),torch.arange(n-1)] = A_off_diag
>>> A
tensor([[ 0.4070+0.4993j, -0.3137-0.5164j,  0.0000+0.0000j,  0.0000+0.0000j],
        [-0.3137-0.5164j, -0.2736+0.1860j, -0.2956-0.1092j,  0.0000+0.0000j],
        [ 0.0000+0.0000j, -0.2956-0.1092j,  0.4033-0.3862j, -0.0085+0.1981j],
        [ 0.0000+0.0000j,  0.0000+0.0000j, -0.0085+0.1981j, -0.1929-0.0194j]])
>>> B = torch.rand(n,k,dtype=torch.complex128,generator=rng)
>>> X_true = torch.linalg.solve(A,B)
>>> X_true
tensor([[-0.0284-1.1816j, -0.6400-0.1975j,  0.7656-1.1306j],
        [-1.6586-0.3570j, -2.4720-0.1880j, -1.1169-0.6640j],
        [-2.4585-1.2378j, -1.0485-2.0358j, -3.1323-0.3632j],
        [ 0.1591-7.4823j,  1.8120-2.6001j, -2.3910-7.6943j]])
>>> torch.allclose(A@X_true-B,torch.zeros_like(B))
True
>>> def A_mult(x):
...     y = x*A_diag[:,None]
...     y[1:,:] += x[:-1,:]*A_off_diag[:,None]
...     y[:-1,:] += x[1:,:]*A_off_diag[:,None]
...     return y
>>> torch.allclose(A_mult(X_true),A@X_true)
True
>>> X_minres = minres_qlp_cs(A_mult,B,verbose=None,verbose_times=False)
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    1          | 7.8e-01   | 8.3e-01   | 8.9e-01   | 9.4e-01   | 9.7e-01   
    2          | 5.1e-01   | 5.2e-01   | 5.2e-01   | 5.4e-01   | 5.5e-01   
    3          | 3.2e-01   | 3.2e-01   | 3.2e-01   | 3.2e-01   | 3.3e-01   
    4          | 9.1e-16   | 1.1e-15   | 1.3e-15   | 2.1e-15   | 2.6e-15   
>>> torch.allclose(X_minres,X_true)
True

Batched tri-diagonal \(A\) with storage-saving multiplication function 

>>> n = 100
>>> k = 3
>>> A_diag = torch.randn(2,1,4,n,dtype=torch.complex128,generator=rng)
>>> A_off_diag = torch.randn(2,1,4,n-1,dtype=torch.complex128,generator=rng) 
>>> A = torch.zeros(2,1,4,n,n,dtype=torch.complex128)
>>> A[...,torch.arange(n),torch.arange(n)] = A_diag 
>>> A[...,torch.arange(n-1),torch.arange(1,n)] = A_off_diag
>>> A[...,torch.arange(1,n),torch.arange(n-1)] = A_off_diag
>>> B = torch.rand(2,6,1,n,k,dtype=torch.complex128,generator=rng)
>>> X_true = torch.linalg.solve(A,B)
>>> torch.allclose(torch.einsum("...ij,...jk->...ik",A,X_true)-B,torch.zeros_like(B))
True
>>> def A_mult(x):
...     y = x*A_diag[...,:,None]
...     y[...,1:,:] += x[...,:-1,:]*A_off_diag[...,:,None]
...     y[...,:-1,:] += x[...,1:,:]*A_off_diag[...,:,None]
...     return y
>>> torch.allclose(A_mult(X_true),torch.einsum("...ij,...jk->...ik",A,X_true))
True
>>> X_minres,data = minres_qlp_cs(A_mult,B,verbose=None,verbose_times=False,store_data_iters=None,store_all_data=True,iters=40)
    iter i     | losses_quantiles                                          
               | 5         | 25        | 50        | 75        | 90        
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
    2          | 6.6e-01   | 6.8e-01   | 7.0e-01   | 7.2e-01   | 7.3e-01   
    4          | 5.1e-01   | 5.4e-01   | 5.6e-01   | 5.8e-01   | 5.9e-01   
    6          | 4.2e-01   | 4.5e-01   | 4.6e-01   | 4.9e-01   | 5.0e-01   
    8          | 3.5e-01   | 3.8e-01   | 4.0e-01   | 4.2e-01   | 4.4e-01   
    10         | 3.0e-01   | 3.3e-01   | 3.6e-01   | 3.8e-01   | 4.0e-01   
    12         | 2.6e-01   | 2.9e-01   | 3.2e-01   | 3.4e-01   | 3.6e-01   
    14         | 2.3e-01   | 2.6e-01   | 2.9e-01   | 3.1e-01   | 3.3e-01   
    16         | 2.1e-01   | 2.3e-01   | 2.6e-01   | 2.8e-01   | 3.0e-01   
    18         | 1.9e-01   | 2.1e-01   | 2.4e-01   | 2.6e-01   | 2.8e-01   
    20         | 1.7e-01   | 1.9e-01   | 2.2e-01   | 2.3e-01   | 2.6e-01   
    22         | 1.5e-01   | 1.7e-01   | 2.0e-01   | 2.2e-01   | 2.4e-01   
    24         | 1.3e-01   | 1.6e-01   | 1.8e-01   | 2.0e-01   | 2.2e-01   
    26         | 1.2e-01   | 1.4e-01   | 1.7e-01   | 1.9e-01   | 2.1e-01   
    28         | 1.0e-01   | 1.3e-01   | 1.6e-01   | 1.8e-01   | 2.0e-01   
    30         | 9.3e-02   | 1.2e-01   | 1.4e-01   | 1.7e-01   | 1.9e-01   
    32         | 8.4e-02   | 1.1e-01   | 1.3e-01   | 1.6e-01   | 1.9e-01   
    34         | 7.6e-02   | 1.0e-01   | 1.2e-01   | 1.6e-01   | 1.8e-01   
    36         | 6.9e-02   | 9.3e-02   | 1.1e-01   | 1.5e-01   | 1.7e-01   
    38         | 6.4e-02   | 8.5e-02   | 1.0e-01   | 1.4e-01   | 1.7e-01   
    40         | 5.8e-02   | 7.5e-02   | 9.2e-02   | 1.4e-01   | 1.6e-01   
>>> X_minres.shape
torch.Size([2, 6, 4, 100, 3])
>>> torch.allclose(X_minres,X_true)
False
>>> print_data_signatures(data)
    data['x'].shape = (2, 6, 4, 100, 3)
    data['iterrange'].shape = (41,)
    data['times'].shape = (41,)
    data['losses_quantiles']
        data['losses_quantiles']['0'].shape = (41,)
        data['losses_quantiles']['1'].shape = (41,)
        data['losses_quantiles']['5'].shape = (41,)
        data['losses_quantiles']['10'].shape = (41,)
        data['losses_quantiles']['25'].shape = (41,)
        data['losses_quantiles']['40'].shape = (41,)
        data['losses_quantiles']['50'].shape = (41,)
        data['losses_quantiles']['60'].shape = (41,)
        data['losses_quantiles']['75'].shape = (41,)
        data['losses_quantiles']['90'].shape = (41,)
        data['losses_quantiles']['95'].shape = (41,)
        data['losses_quantiles']['99'].shape = (41,)
        data['losses_quantiles']['100'].shape = (41,)
    data['xs'].shape = (41, 2, 6, 4, 100, 3)
    data['losses'].shape = (41, 2, 6, 4, 3)
Source code in agsutil/algos.py 
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def minres_qlp_cs(
        A,
        B,
        X0 = None,
        iters = None,
        residtol = None,
        verbose = False, 
        verbose_indent = 4,
        quantiles_losses = [0,1,5,10,25,40,50,60,75,90,95,99,100],
        verbose_quantiles_losses = [5,25,50,75,90],
        verbose_times = True, 
        warn = True,
        store_data_iters = False, 
        store_all_data = False,
        ):
    # https://github.com/schoi32/sci498rms/blob/master/Sorokin_MinresQLP_Python_Workspace/MinresQLP/Algorithms/cs_mqlp.py
    r"""
    MINRES QLP algorith for complex-symmetric matrices. 

    A translation of the [MATLAB version of MINRESQLP](http://www.stanford.edu/group/SOL/software.html).

    References 

    1.  S.-C. Choi, C. C. Paige, and M. A. Saunders,
        MINRES-QLP: A Krylov subspace method for indefinite or singular symmetric systems,
        SIAM Journal of Scientific Computing, submitted on March 7, 2010.

    2.  S.-C. Choi's PhD Dissertation, Stanford University, 2006: 
        http://www.stanford.edu/group/SOL/dissertations.html

    Args:
        A (Union[torch.Tensor,callable]): Symmetric matrix `A` with shape `(...,n,n)`, or  
            `callable(A)` where `a(x)` should return the batch matrix multiplication of `A` and `X`,  
        B (torch.Tensor): Right hand side tensor $B$ with shape `(...,n,k)`
        X0 (torch.Tensor): Initial guess for $X$ with shape `(...,n,k)`, defaults to zeros. 
        iters (int): number of minres iterations, defaults to `5n`. 
        residtol (float): Non-negative tolerance on the maximum residual for early stopping, defaults to `1e-12` for `torch.float64` and `2.5e-4` for `torch.float32`.
        verbose (int): Controls logging verbosity

            - If `True`, perform logging. 
            - If a positive int, only log every `verbose` iterations. 
            - If `None`, set to a reasonable positive int based on the maximum number of iterations
            - If `False`, don't log. 

        verbose_indent (int): Non-negative number of indentation spaces for logging.
        quantiles_losses (list): Loss quantiles to record.
        verbose_quantiles_losses (list): Loss quantiles to show in verbose log.
        verbose_times (bool): If `False`, do not show the times in the verbose log. This is mostly for testing where timing is not reproducible. 
        warn (bool): If `False`, then suppress warnings.
        store_data_iters (int): Controls storage iterations with the same options as verbose. If `store_data_iters==0`, then the data is not collected or returned. 

            - If `True`, store every iteration. 
            - If a positive int, only store every `store_data_iters` iterations. 
            - If `None`, set to a reasonable positive int based on the maximum number of iterations
            - If `False`, don't store data, and do not return data 

        store_all_data (bool): If `True`, store the `x` values as well as the metrics. 

    Returns:
        x (torch.Tensor): Optimized $X$.
        data (dict): Iteration data, only returned when `store_data_iters>0`

    Examples:

        >>> torch.set_default_dtype(torch.float64)
        >>> rng = torch.Generator().manual_seed(7)

    Column vector $b$ 

        >>> n = 5
        >>> A = torch.randn(n,n,dtype=torch.complex128,generator=rng)
        >>> A = (A+A.T)/2
        >>> b = torch.rand(n,dtype=torch.complex128,generator=rng)
        >>> x_true = torch.linalg.solve(A,b[...,None])[...,0]
        >>> x_true
        tensor([-0.6207-0.4121j,  0.5221+0.3249j, -1.0952+0.8594j,  0.9080-1.2110j,
                -0.9799+0.7372j])
        >>> torch.allclose(A@x_true-b,torch.zeros_like(b))
        True
        >>> x_minres = minres_qlp_cs(A,b[...,None],verbose=None,verbose_times=False)[...,0]
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 9.6e-01   | 9.6e-01   | 9.6e-01   | 9.6e-01   | 9.6e-01   
            2          | 5.8e-01   | 5.8e-01   | 5.8e-01   | 5.8e-01   | 5.8e-01   
            3          | 4.5e-01   | 4.5e-01   | 4.5e-01   | 4.5e-01   | 4.5e-01   
            4          | 2.8e-01   | 2.8e-01   | 2.8e-01   | 2.8e-01   | 2.8e-01   
            5          | 1.7e-14   | 1.7e-14   | 1.7e-14   | 1.7e-14   | 1.7e-14   
        >>> torch.allclose(x_minres,x_true)
        True

    Matrix $B$

        >>> n = 5
        >>> k = 3
        >>> A = torch.randn(n,n,dtype=torch.complex128,generator=rng)
        >>> A = (A+A.T)/2
        >>> B = torch.rand(n,k,dtype=torch.complex128,generator=rng)
        >>> X_true = torch.linalg.solve(A,B)
        >>> X_true
        tensor([[ 0.0142+0.7190j,  0.0097+0.9734j, -0.3620+0.6413j],
                [ 0.4527+0.7455j,  0.4270+0.4941j,  0.9255+0.5685j],
                [-0.7182-1.0284j,  0.3193-0.4463j, -0.3421-0.8780j],
                [-0.5973+0.4910j, -0.8147+0.8256j, -0.8821+0.7147j],
                [-0.0926+0.5500j, -0.5192+0.2817j, -0.8496+0.6976j]])
        >>> torch.allclose(A@X_true-B,torch.zeros_like(B))
        True
        >>> X_minres = minres_qlp_cs(A,B,verbose=None,verbose_times=False)
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 9.0e-01   | 9.1e-01   | 9.2e-01   | 9.5e-01   | 9.7e-01   
            2          | 2.2e-01   | 3.1e-01   | 4.3e-01   | 5.0e-01   | 5.5e-01   
            3          | 2.0e-01   | 2.2e-01   | 2.3e-01   | 2.9e-01   | 3.3e-01   
            4          | 3.4e-02   | 5.0e-02   | 7.1e-02   | 8.1e-02   | 8.7e-02   
            5          | 1.9e-15   | 4.2e-15   | 7.0e-15   | 1.5e-14   | 2.0e-14   
        >>> torch.allclose(X_minres,X_true)
        True

    Tri-diagonal $A$ with storage-saving multiplication function 

        >>> n = 4
        >>> k = 3
        >>> A_diag = torch.randn(n,dtype=torch.complex128,generator=rng)
        >>> A_off_diag = torch.randn(n-1,dtype=torch.complex128,generator=rng) 
        >>> A = torch.zeros(n,n,dtype=torch.complex128)
        >>> A[torch.arange(n),torch.arange(n)] = A_diag 
        >>> A[torch.arange(n-1),torch.arange(1,n)] = A_off_diag
        >>> A[torch.arange(1,n),torch.arange(n-1)] = A_off_diag
        >>> A
        tensor([[ 0.4070+0.4993j, -0.3137-0.5164j,  0.0000+0.0000j,  0.0000+0.0000j],
                [-0.3137-0.5164j, -0.2736+0.1860j, -0.2956-0.1092j,  0.0000+0.0000j],
                [ 0.0000+0.0000j, -0.2956-0.1092j,  0.4033-0.3862j, -0.0085+0.1981j],
                [ 0.0000+0.0000j,  0.0000+0.0000j, -0.0085+0.1981j, -0.1929-0.0194j]])
        >>> B = torch.rand(n,k,dtype=torch.complex128,generator=rng)
        >>> X_true = torch.linalg.solve(A,B)
        >>> X_true
        tensor([[-0.0284-1.1816j, -0.6400-0.1975j,  0.7656-1.1306j],
                [-1.6586-0.3570j, -2.4720-0.1880j, -1.1169-0.6640j],
                [-2.4585-1.2378j, -1.0485-2.0358j, -3.1323-0.3632j],
                [ 0.1591-7.4823j,  1.8120-2.6001j, -2.3910-7.6943j]])
        >>> torch.allclose(A@X_true-B,torch.zeros_like(B))
        True
        >>> def A_mult(x):
        ...     y = x*A_diag[:,None]
        ...     y[1:,:] += x[:-1,:]*A_off_diag[:,None]
        ...     y[:-1,:] += x[1:,:]*A_off_diag[:,None]
        ...     return y
        >>> torch.allclose(A_mult(X_true),A@X_true)
        True
        >>> X_minres = minres_qlp_cs(A_mult,B,verbose=None,verbose_times=False)
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            1          | 7.8e-01   | 8.3e-01   | 8.9e-01   | 9.4e-01   | 9.7e-01   
            2          | 5.1e-01   | 5.2e-01   | 5.2e-01   | 5.4e-01   | 5.5e-01   
            3          | 3.2e-01   | 3.2e-01   | 3.2e-01   | 3.2e-01   | 3.3e-01   
            4          | 9.1e-16   | 1.1e-15   | 1.3e-15   | 2.1e-15   | 2.6e-15   
        >>> torch.allclose(X_minres,X_true)
        True

    Batched tri-diagonal $A$ with storage-saving multiplication function 

        >>> n = 100
        >>> k = 3
        >>> A_diag = torch.randn(2,1,4,n,dtype=torch.complex128,generator=rng)
        >>> A_off_diag = torch.randn(2,1,4,n-1,dtype=torch.complex128,generator=rng) 
        >>> A = torch.zeros(2,1,4,n,n,dtype=torch.complex128)
        >>> A[...,torch.arange(n),torch.arange(n)] = A_diag 
        >>> A[...,torch.arange(n-1),torch.arange(1,n)] = A_off_diag
        >>> A[...,torch.arange(1,n),torch.arange(n-1)] = A_off_diag
        >>> B = torch.rand(2,6,1,n,k,dtype=torch.complex128,generator=rng)
        >>> X_true = torch.linalg.solve(A,B)
        >>> torch.allclose(torch.einsum("...ij,...jk->...ik",A,X_true)-B,torch.zeros_like(B))
        True
        >>> def A_mult(x):
        ...     y = x*A_diag[...,:,None]
        ...     y[...,1:,:] += x[...,:-1,:]*A_off_diag[...,:,None]
        ...     y[...,:-1,:] += x[...,1:,:]*A_off_diag[...,:,None]
        ...     return y
        >>> torch.allclose(A_mult(X_true),torch.einsum("...ij,...jk->...ik",A,X_true))
        True
        >>> X_minres,data = minres_qlp_cs(A_mult,B,verbose=None,verbose_times=False,store_data_iters=None,store_all_data=True,iters=40)
            iter i     | losses_quantiles                                          
                       | 5         | 25        | 50        | 75        | 90        
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            0          | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   | 1.0e+00   
            2          | 6.6e-01   | 6.8e-01   | 7.0e-01   | 7.2e-01   | 7.3e-01   
            4          | 5.1e-01   | 5.4e-01   | 5.6e-01   | 5.8e-01   | 5.9e-01   
            6          | 4.2e-01   | 4.5e-01   | 4.6e-01   | 4.9e-01   | 5.0e-01   
            8          | 3.5e-01   | 3.8e-01   | 4.0e-01   | 4.2e-01   | 4.4e-01   
            10         | 3.0e-01   | 3.3e-01   | 3.6e-01   | 3.8e-01   | 4.0e-01   
            12         | 2.6e-01   | 2.9e-01   | 3.2e-01   | 3.4e-01   | 3.6e-01   
            14         | 2.3e-01   | 2.6e-01   | 2.9e-01   | 3.1e-01   | 3.3e-01   
            16         | 2.1e-01   | 2.3e-01   | 2.6e-01   | 2.8e-01   | 3.0e-01   
            18         | 1.9e-01   | 2.1e-01   | 2.4e-01   | 2.6e-01   | 2.8e-01   
            20         | 1.7e-01   | 1.9e-01   | 2.2e-01   | 2.3e-01   | 2.6e-01   
            22         | 1.5e-01   | 1.7e-01   | 2.0e-01   | 2.2e-01   | 2.4e-01   
            24         | 1.3e-01   | 1.6e-01   | 1.8e-01   | 2.0e-01   | 2.2e-01   
            26         | 1.2e-01   | 1.4e-01   | 1.7e-01   | 1.9e-01   | 2.1e-01   
            28         | 1.0e-01   | 1.3e-01   | 1.6e-01   | 1.8e-01   | 2.0e-01   
            30         | 9.3e-02   | 1.2e-01   | 1.4e-01   | 1.7e-01   | 1.9e-01   
            32         | 8.4e-02   | 1.1e-01   | 1.3e-01   | 1.6e-01   | 1.9e-01   
            34         | 7.6e-02   | 1.0e-01   | 1.2e-01   | 1.6e-01   | 1.8e-01   
            36         | 6.9e-02   | 9.3e-02   | 1.1e-01   | 1.5e-01   | 1.7e-01   
            38         | 6.4e-02   | 8.5e-02   | 1.0e-01   | 1.4e-01   | 1.7e-01   
            40         | 5.8e-02   | 7.5e-02   | 9.2e-02   | 1.4e-01   | 1.6e-01   
        >>> X_minres.shape
        torch.Size([2, 6, 4, 100, 3])
        >>> torch.allclose(X_minres,X_true)
        False
        >>> print_data_signatures(data)
            data['x'].shape = (2, 6, 4, 100, 3)
            data['iterrange'].shape = (41,)
            data['times'].shape = (41,)
            data['losses_quantiles']
                data['losses_quantiles']['0'].shape = (41,)
                data['losses_quantiles']['1'].shape = (41,)
                data['losses_quantiles']['5'].shape = (41,)
                data['losses_quantiles']['10'].shape = (41,)
                data['losses_quantiles']['25'].shape = (41,)
                data['losses_quantiles']['40'].shape = (41,)
                data['losses_quantiles']['50'].shape = (41,)
                data['losses_quantiles']['60'].shape = (41,)
                data['losses_quantiles']['75'].shape = (41,)
                data['losses_quantiles']['90'].shape = (41,)
                data['losses_quantiles']['95'].shape = (41,)
                data['losses_quantiles']['99'].shape = (41,)
                data['losses_quantiles']['100'].shape = (41,)
            data['xs'].shape = (41, 2, 6, 4, 100, 3)
            data['losses'].shape = (41, 2, 6, 4, 3)
    """
    if warn and (not torch.get_default_dtype()==torch.float64): warnings.warn('''
            torch.get_default_dtype() = %s, but lm_opt often requires high precision updates. We recommend using:
                torch.set_default_dtype(torch.float64)'''%str(torch.get_default_dtype()))
    assert torch.get_default_dtype() in [torch.float32,torch.float64]
    default_dtype = torch.get_default_dtype()
    device = str(B.device)
    default_device = str(torch.get_default_device())
    assert B.ndim>=2, "B should have shape (...,n,k)"
    n = B.size(-2)
    k = B.size(-1)
    assert B.dtype in [torch.complex64,torch.complex128]
    if X0 is None: 
        X0 = torch.zeros_like(B)
    if isinstance(A,torch.Tensor):
        assert A.shape[-2:]==(n,n)
        assert torch.allclose(A.T,A)
        matvec = lambda X: torch.einsum("...ij,...jk->...ik",A,X)
    else:
        assert callable(A)
        matvec = A
    if iters is None: 
        iters = 5*n 
    assert iters>=0
    assert iters%1==0
    if residtol is None: 
        if default_dtype==torch.float64:
            residtol = 1e-12
        elif default_dtype==torch.float32:
            residtol = 2.5e-4
        else:
            raise Exception("default_dtype = %s not parsed"%str(default_dtype))
    assert residtol>=0
    if verbose is None: 
        verbose = max(1,iters//20)
    assert verbose%1==0
    assert verbose>=0 
    if store_data_iters is None: 
        store_data_iters = max(1,iters//1000)
    assert store_data_iters%1==0
    assert store_data_iters>=0 
    assert isinstance(store_all_data,bool)
    assert isinstance(quantiles_losses,list)
    assert all(0<=qt<=100 for qt in quantiles_losses)
    assert isinstance(verbose_quantiles_losses,list)
    assert all(qt in quantiles_losses for qt in verbose_quantiles_losses)
    assert verbose_indent%1==0 
    assert verbose_indent>=0
    assert isinstance(verbose_times,bool)
    if verbose:
        _h_iter = "%-10s "%"iter i"
        _h_times = "| %-10s"%"times" if verbose_times else ""
        _s_losses_qt = ("| %-9s "*len(verbose_quantiles_losses))%tuple(str(qt) for qt in verbose_quantiles_losses)
        _h_losses_qt = "| losses_quantiles"+" "*(len(_s_losses_qt)-len("| losses_quantiles"))
        _h = _h_iter+_h_losses_qt+_h_times
        _s = " "*len(_h_iter)+_s_losses_qt+("|"+" "*(len(_h_times)-1) if verbose_times else " "*len(_h_times))
        print(" "*verbose_indent+_h)
        print(" "*verbose_indent+_s)
        print(" "*verbose_indent+"~"*len(_s))
    timer = Timer(device=device)
    timer.tic()
    psolve = lambda X: X # TODO: implement more involved preconditioned solver
    inner = lambda a,b: torch.einsum("...ij,...ij->...j",a.conj(),b)
    Anorm = 0
    x = X0 
    Ax = matvec(x)
    assert Ax.shape[-2:]==(n,k)
    batch_shape = tuple(Ax.shape[:-2])
    if store_data_iters:
        iterrange = []
        times = []
        losses = []
        losses_quantiles = {str(qt):[] for qt in quantiles_losses}
        if store_all_data:
            xs = []
    r2 = B  # (...,k)
    r3 = r2 # (...,k)
    r3 = psolve(r2)
    beta1 = torch.sqrt(inner(r3,r2))  # (...,k)
    bnorm = torch.linalg.norm(B,dim=-2) # (...,k)
    # TODO: Check if below variables are necessary    
    beta = torch.zeros_like(beta1)
    tau = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    taul = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    phi = beta1
    betan = beta1
    cs = -torch.ones((*batch_shape,k),dtype=B.dtype,device=device)
    sn = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    cr1 = torch.ones((*batch_shape,k),dtype=B.dtype,device=device)
    sr1 = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    cr2 = -torch.ones((*batch_shape,k),dtype=B.dtype,device=device)
    sr2 = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    dltan = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    eplnn = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    gama = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    gamal = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    gamal2 = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    eta = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    etal = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    etal2 = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    vepln = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    veplnl = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    veplnl2 = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    ul3 = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    ul2 = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    ul = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    u = torch.zeros((*batch_shape,k),dtype=B.dtype,device=device)
    w = torch.zeros_like(B)
    wl = torch.zeros_like(B)
    r1 = torch.zeros_like(B)
    xl2 = torch.zeros_like(B)
    alfa = torch.zeros_like(B)
    shift = 0 # TODO: If shift != 0 then the method solves (A - shift*I)x = b
    for i in range(iters+1):
        resid = matvec(x)-B 
        breakcond = i==iters or resid.abs().amax()<=residtol
        loss = torch.linalg.norm(resid,dim=-2)/bnorm
        times_i = timer.toc()
        losses_quantiles_i = {str(qt): loss.nanquantile(qt/100) for qt in quantiles_losses}
        if store_data_iters and (i%store_data_iters==0 or breakcond):
            iterrange.append(i)
            losses.append(loss.to(default_device))
            times.append(times_i)
            for qt in quantiles_losses:
                losses_quantiles[str(qt)].append(losses_quantiles_i[str(qt)].to(default_device))
            if store_all_data:
                xs.append(x.expand(resid.shape).to(default_device))
        if verbose and (i%verbose==0 or breakcond):
            _s_iter = "%-10d "%i
            _s_losses_qt = ("| %-9.1e "*len(verbose_quantiles_losses))%tuple(losses_quantiles_i[str(qt)] for qt in verbose_quantiles_losses)
            _s_times = "| %-10.1f "%(times_i) if verbose_times else ""
            print(" "*verbose_indent+_s_iter+_s_losses_qt+_s_times)
        if breakcond: break 
        betal = beta
        beta = betan
        v = r3/beta[...,None,:]    
        r3 = matvec(v.conj())-shift*v.conj()
        if i>0:
            r3 = r3-(beta/betal)[...,None,:]*r1
        alfa = inner(v,r3)
        r3 = r3-(alfa/beta)[...,None,:]*r2
        r1 = r2
        r2 = r3
        r3 = psolve(r2)
        betan = torch.sqrt(inner(r2,r3))
        dbar = dltan
        dlta = cs*dbar+sn*alfa
        epln = eplnn
        gbar = sn.conj()*dbar-cs*alfa
        eplnn = sn*betan
        dltan = -cs*betan
        dlta_QLP = dlta
        gamal3 = gamal2
        gamal2 = gamal
        gamal = gama
        (cs,sn,gama) = symOrtho(gbar,betan)
        gama_tmp = gama
        taul2  = taul
        taul = tau
        tau = cs*phi
        phi = sn.conj()*phi
        if i>1:
            veplnl2  = veplnl
            etal2 = etal
            etal = eta
            dlta_tmp = sr2*vepln - cr2*dlta
            veplnl = cr2*vepln+sr2.conj()*dlta
            dlta = dlta_tmp
            eta = sr2.conj()*gama
            gama = -cr2*gama
        if i>0:
            (cr1,sr1,gamal) = symOrtho(gamal.conj(),dlta.conj())
            gamal = gamal.conj()
            vepln = sr1.conj()*gama
            gama = -cr1*gama
        ul4 = ul3
        ul3 = ul2
        if i>1:
            ul2 = (taul2-etal2*ul4-veplnl2*ul3)/gamal2
        if i>0:
            ul = (taul-etal*ul3-veplnl*ul2)/gamal
        u = (tau - eta*ul2 - vepln*ul) / gama
        if i==0:
            wl2 = wl
            wl = v.conj()*sr1.conj()[...,None,:]
            w  = v.conj()*cr1[...,None,:]
        elif i==1:
            wl2 = wl
            wl = w*cr1[...,None,:]+v.conj()*sr1.conj()[...,None,:]
            w = w*sr1[...,None,:]-v.conj()*cr1[...,None,:]
        else:
            wl2 = wl
            wl = w
            w  = wl2*sr2[...,None,:]-v.conj()*cr2[...,None,:]
            wl2 = wl2*cr2[...,None,:]+v.conj()*sr2.conj()[...,None,:]
            v = wl*cr1[...,None,:]+w*sr1.conj()[...,None,:]
            w = wl*sr1[...,None,:]-w*cr1[...,None,:]
            wl = v
        xl2 = xl2+wl2*ul2[...,None,:]
        x = xl2+wl*ul[...,None,:]+w*u[...,None,:]
        pass
        (cr2,sr2,gamal) = symOrtho(gamal.conj(),eplnn.conj())
        gamal = gamal.conj()
    if store_data_iters==0:
        return x 
    else:
        data = {
            "x": x.to(default_device), 
            "iterrange": torch.tensor(iterrange,dtype=int), 
            "times": torch.tensor(times), 
            "losses_quantiles": {str(qt):torch.tensor(losses_quantiles[str(qt)]) for qt in quantiles_losses},
            }
        if store_all_data:
            data["xs"] = torch.stack(xs,dim=0)
            data["losses"] = torch.stack(losses,dim=0)
        return x,data

Misc utils

Timer 

Timer(device)

Timer compatible with CPU and GPU operations.

Parameters:

Name Type Description Default
device Union[device, str]

device to perform timing for.

  • For CPU devices, time.perf_counter() is used.
  • For CUDA and MPS GPU devices, torch.{cuda,mps}.Event(enable_timing=True) is used.
 required
Source code in agsutil/utils.py 
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def __init__(self, device):
    r"""
    Args:
        device (Union[torch.device,str]): device to perform timing for. 

            - For CPU devices, `time.perf_counter()` is used. 
            - For CUDA and MPS GPU devices, `torch.{cuda,mps}.Event(enable_timing=True)` is used. 
    """

    device_str = str(device) 
    if "cpu" in device_str:
        self.torch_backend = torch.cpu 
    elif "cuda" in device_str:
        self.torch_backend = torch.cuda 
    elif "mps" in device_str:
        self.torch_backend = torch.mps 
    else:
        raise Exception("undetected device = %s, should have 'cpu', 'cuda', or 'mps' in it."%device_str)

tic 

tic()

Start the stopwatch.

Source code in agsutil/utils.py 
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def tic(self):
    r"""
    Start the stopwatch.
    """
    if self.torch_backend==torch.cpu:
        self.t0 = time.perf_counter()
    else:
        self.torch_backend.empty_cache()
        self.t0 = self.torch_backend.Event(enable_timing=True)
        self.tend = self.torch_backend.Event(enable_timing=True)
        self.t0.record()

toc 

toc()

Lap the stopwatch.

Returns:

Name Type Description
tdelta float

time elapsed between the start of the stopwatch and the current lap.
Source code in agsutil/utils.py 
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def toc(self):
    r"""
    Lap the stopwatch. 

    Returns: 
        tdelta (float): time elapsed between the start of the stopwatch and the current lap.
    """
    if self.torch_backend==torch.cpu:
        tdelta = time.perf_counter()-self.t0
    else:
        self.tend.record()
        self.torch_backend.synchronize()
        tdelta = self.t0.elapsed_time(self.tend)/1000
    return float(tdelta)

print_data_signatures 

print_data_signatures(data, name='data', print_devices=False, print_dtypes=False, verbose_indent=4)

Print data shapes and (optionally) devices.

Parameters:

Name Type Description Default
data dict

Dictiony with items that are tensors or dictionaries of tensors.

required
print_devices bool

If True, also print the device.

False
print_dtypes bool

If True, also print the dtypes.

False
verbose_indent int

Non-negative number of indentation spaces for logging.

 4

Examples:

>>> data = {
...     "a": torch.rand(2,3,4),
...     "b": torch.rand(3,4,5),
...     "subdata": {
...         "aa": torch.rand(2,3),
...         "bb": torch.rand(2,3),
...         "subnontensor": ["ags",7,7,7],
...         },
...     "nontensor": [7,7,7,"ags"]
...     }
>>> print_data_signatures(data,print_devices=True,print_dtypes=True,verbose_indent=0)
data['a'].shape = (2, 3, 4) on device = cpu with dtype = torch.float64
data['b'].shape = (3, 4, 5) on device = cpu with dtype = torch.float64
data['subdata']
    data['subdata']['aa'].shape = (2, 3) on device = cpu with dtype = torch.float64
    data['subdata']['bb'].shape = (2, 3) on device = cpu with dtype = torch.float64
    data['subdata']['subnontensor'] a list of length 4
data['nontensor'] a list of length 4
Source code in agsutil/utils.py 
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def print_data_signatures(data, name="data", print_devices=False, print_dtypes=False, verbose_indent=4):
    r""" 
    Print data shapes and (optionally) devices. 

    Args: 
        data (dict): Dictiony with items that are tensors or dictionaries of tensors. 
        print_devices (bool): If `True`, also print the device. 
        print_dtypes (bool): If `True`, also print the dtypes. 
        verbose_indent (int): Non-negative number of indentation spaces for logging.

    Examples:
        >>> data = {
        ...     "a": torch.rand(2,3,4),
        ...     "b": torch.rand(3,4,5),
        ...     "subdata": {
        ...         "aa": torch.rand(2,3),
        ...         "bb": torch.rand(2,3),
        ...         "subnontensor": ["ags",7,7,7],
        ...         },
        ...     "nontensor": [7,7,7,"ags"]
        ...     }
        >>> print_data_signatures(data,print_devices=True,print_dtypes=True,verbose_indent=0)
        data['a'].shape = (2, 3, 4) on device = cpu with dtype = torch.float64
        data['b'].shape = (3, 4, 5) on device = cpu with dtype = torch.float64
        data['subdata']
            data['subdata']['aa'].shape = (2, 3) on device = cpu with dtype = torch.float64
            data['subdata']['bb'].shape = (2, 3) on device = cpu with dtype = torch.float64
            data['subdata']['subnontensor'] a list of length 4
        data['nontensor'] a list of length 4
    """ 
    for key,val in data.items():
        if isinstance(val,torch.Tensor):
            _s = "%s['%s'].shape = %s"%(name,key,str(tuple(data[key].shape)))
            if print_devices:
                _s += " on device = %s"%str(data[key].device)
            if print_dtypes:
                _s += " with dtype = %s"%str(data[key].dtype)
            print(" "*verbose_indent+_s)
        elif isinstance(val,dict):
            print(" "*verbose_indent+"data['%s']"%key)
            for kkey,vval in val.items():
                if isinstance(vval,torch.Tensor):
                    _s = "%s['%s']['%s'].shape = %s"%(name,key,kkey,str(tuple(data[key][kkey].shape)))
                    if print_devices:
                        _s += " on device = %s"%str(data[key][kkey].device)
                    if print_dtypes:
                        _s += " with dtype = %s"%str(data[key][kkey].dtype)
                    print(" "*(verbose_indent+4)+_s)
                elif isinstance(vval,list):
                    print(" "*(verbose_indent+4)+"%s['%s']['%s'] a list of length %d"%(name,key,kkey,len(data[key][kkey])))
                else:
                    print(" "*(verbose_indent+4)+"%s['%s']['%s'] = %s"%(name,key,kkey,str(data[key][kkey])))
        elif isinstance(val,list):
            print(" "*verbose_indent+"%s['%s'] a list of length %d"%(name,key,len(data[key])))
        else:
            print(" "*verbose_indent+"%s['%s'] = %s"%(name,key,str(data[key])))

to_unitary_expskewh 

to_unitary_expskewh(theta, n, complex_case=False)

Transform to a unitary matrix using the exponential of a skew Hermitian matrix.

Parameters:

Name Type Description Default
theta Tensor

With theta.size(-1) == n*(n-1)//2 in the real case and theta.size(-1) == n**2 in the complex case.

 required
complex_case bool

If True, parameterize a complex matrix, otherwise a real one.

False

Returns:

Name Type Description
Q Tensor

With shape (*theta.shape[:-1],n,n) unitary matrices

Examples:

>>> torch.set_default_dtype(torch.float64)
>>> rng = torch.Generator().manual_seed(7)

Single matrix

>>> n = 5
>>> theta = torch.rand(n*(n-1)//2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n)
>>> Q 
tensor([[ 0.5027, -0.1038, -0.2508,  0.3769,  0.7291],
        [-0.6870,  0.3679,  0.2777,  0.3454,  0.4430],
        [-0.2330, -0.7975,  0.3425, -0.2999,  0.3201],
        [-0.4638, -0.3141, -0.8037,  0.1779, -0.0934],
        [-0.0769,  0.3453, -0.3111, -0.7855,  0.4013]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
True

Single complex matrix

>>> n = 4
>>> theta = torch.rand(n**2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n,complex_case=True)
>>> Q 
tensor([[ 0.1358-0.3038j, -0.4798+0.2431j,  0.1105+0.2308j, -0.2229+0.6963j],
        [-0.4395+0.6202j,  0.1556-0.1974j, -0.1838-0.0838j, -0.1181+0.5516j],
        [-0.4750+0.0762j, -0.7622+0.0020j, -0.2319-0.0263j,  0.2346-0.2795j],
        [-0.0745-0.2729j,  0.2531+0.0496j, -0.7624+0.5079j,  0.1119+0.0410j]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
True

Two matrices

>>> n = 3
>>> theta = torch.rand(2,n*(n-1)//2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n)
>>> Q 
tensor([[[ 0.3004,  0.6612,  0.6874],
         [-0.7630,  0.5990, -0.2428],
         [-0.5723, -0.4516,  0.6845]],
<BLANKLINE>
        [[ 0.4583,  0.7552,  0.4687],
         [-0.7977,  0.5820, -0.1579],
         [-0.3921, -0.3015,  0.8691]]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
True

Two complex matrices

>>> n = 3
>>> theta = torch.rand(2,n**2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n,complex_case=True)
>>> Q 
tensor([[[ 0.5591+0.3745j, -0.4342+0.5076j, -0.1283+0.2907j],
         [-0.4802+0.4013j,  0.5099+0.4179j, -0.2657+0.3211j],
         [-0.3843+0.0887j, -0.2972+0.1756j,  0.8185+0.2353j]],
<BLANKLINE>
        [[ 0.4860+0.5953j, -0.1642+0.0458j, -0.3162+0.5295j],
         [-0.5638+0.0805j,  0.3753+0.0607j,  0.2470+0.6856j],
         [-0.2898+0.0325j, -0.8621+0.2884j,  0.2790+0.1037j]]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
True

Batch support

>>> n = 10
>>> theta = torch.rand(2,3,4,n*(n-1)//2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n)
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
True
>>> theta = torch.rand(2,3,4,n**2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n,complex_case=True)
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
True
Source code in agsutil/utils.py 
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def to_unitary_expskewh(theta, n, complex_case=False):
    r"""
    Transform to a unitary matrix using the exponential of a skew Hermitian matrix.

    Args:
        theta (torch.Tensor): With `theta.size(-1) == n*(n-1)//2` in the real case and `theta.size(-1) == n**2` in the complex case.
        complex_case (bool): If `True`, parameterize a complex matrix, otherwise a real one. 

    Returns:
        Q (torch.Tensor): With shape `(*theta.shape[:-1],n,n)` unitary matrices

    Examples:
        >>> torch.set_default_dtype(torch.float64)
        >>> rng = torch.Generator().manual_seed(7)

    Single matrix

        >>> n = 5
        >>> theta = torch.rand(n*(n-1)//2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n)
        >>> Q 
        tensor([[ 0.5027, -0.1038, -0.2508,  0.3769,  0.7291],
                [-0.6870,  0.3679,  0.2777,  0.3454,  0.4430],
                [-0.2330, -0.7975,  0.3425, -0.2999,  0.3201],
                [-0.4638, -0.3141, -0.8037,  0.1779, -0.0934],
                [-0.0769,  0.3453, -0.3111, -0.7855,  0.4013]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
        True

    Single complex matrix

        >>> n = 4
        >>> theta = torch.rand(n**2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n,complex_case=True)
        >>> Q 
        tensor([[ 0.1358-0.3038j, -0.4798+0.2431j,  0.1105+0.2308j, -0.2229+0.6963j],
                [-0.4395+0.6202j,  0.1556-0.1974j, -0.1838-0.0838j, -0.1181+0.5516j],
                [-0.4750+0.0762j, -0.7622+0.0020j, -0.2319-0.0263j,  0.2346-0.2795j],
                [-0.0745-0.2729j,  0.2531+0.0496j, -0.7624+0.5079j,  0.1119+0.0410j]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
        True

    Two matrices

        >>> n = 3
        >>> theta = torch.rand(2,n*(n-1)//2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n)
        >>> Q 
        tensor([[[ 0.3004,  0.6612,  0.6874],
                 [-0.7630,  0.5990, -0.2428],
                 [-0.5723, -0.4516,  0.6845]],
        <BLANKLINE>
                [[ 0.4583,  0.7552,  0.4687],
                 [-0.7977,  0.5820, -0.1579],
                 [-0.3921, -0.3015,  0.8691]]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
        True

    Two complex matrices

        >>> n = 3
        >>> theta = torch.rand(2,n**2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n,complex_case=True)
        >>> Q 
        tensor([[[ 0.5591+0.3745j, -0.4342+0.5076j, -0.1283+0.2907j],
                 [-0.4802+0.4013j,  0.5099+0.4179j, -0.2657+0.3211j],
                 [-0.3843+0.0887j, -0.2972+0.1756j,  0.8185+0.2353j]],
        <BLANKLINE>
                [[ 0.4860+0.5953j, -0.1642+0.0458j, -0.3162+0.5295j],
                 [-0.5638+0.0805j,  0.3753+0.0607j,  0.2470+0.6856j],
                 [-0.2898+0.0325j, -0.8621+0.2884j,  0.2790+0.1037j]]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
        True

    Batch support

        >>> n = 10
        >>> theta = torch.rand(2,3,4,n*(n-1)//2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n)
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
        True
        >>> theta = torch.rand(2,3,4,n**2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n,complex_case=True)
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
        True
    """
    batch_shape = tuple(theta.shape[:-1])
    batch_ones = torch.ones(batch_shape,dtype=int,device=theta.device)
    iut = torch.triu_indices(n,n,offset=1,device=theta.device)
    iutf = torch.einsum("...,i->...i",batch_ones,iut[0]*n+iut[1])
    iltf = torch.einsum("...,i->...i",batch_ones,iut[1]*n+iut[0])
    if complex_case:
        assert theta.size(-1)==(n**2)
        complex_dtype = (theta+0j).dtype
        H = torch.zeros((*batch_shape,n*n),dtype=complex_dtype,device=theta.device)
        idiag = torch.arange(n,device=theta.device)
        idiagf = torch.einsum("...,i->...i",batch_ones,idiag*n+idiag)
        diag = 1j*theta[...,:n].to(complex_dtype)
        off_real,off_imag = theta[...,n:n+(n**2-n)//2],theta[...,n+(n**2-n)//2:]
        off_vals = off_real.to(complex_dtype)+1j*off_imag.to(complex_dtype)
        H = H.scatter_add(-1,idiagf,diag)
        H = H.scatter_add(-1,iutf,off_vals)
        H = H.scatter_add(-1,iltf,-off_vals.conj())
    else:
        assert theta.size(-1)==(n*(n-1)//2)
        H = torch.zeros((*batch_shape,n*n),dtype=theta.dtype,device=theta.device)
        H = H.scatter_add(-1,iutf,theta)
        H = H.scatter_add(-1,iltf,-theta)
    H = H.reshape((*batch_shape,n,n))
    return torch.matrix_exp(H)

from_unitary_expskewh 

from_unitary_expskewh(Q, complex_case=False)

Transform from a unitary matrix using the exponential of a skew Hermitian matrix.

Parameters:

Name Type Description Default
Q Tensor

With shape (*theta.shape[:-1],n,n) unitary matrices

 required
complex_case bool

If True, parameterize a complex matrix, otherwise a real one.

False

Returns:

Name Type Description
theta Tensor

With theta.size(-1) == n*(n-1)//2 in the real case and theta.size(-1) == n**2 in the complex case.
Note that this theta is not unique, as shown in the following doctests.

Examples:

>>> torch.set_default_dtype(torch.float64)
>>> rng = torch.Generator().manual_seed(7)

Single matrix

>>> n = 10
>>> theta = torch.rand(n*(n-1)//2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n)
>>> theta2 = from_unitary_expskewh(Q)
>>> torch.allclose(theta,theta2)
False
>>> Q2 = to_unitary_expskewh(theta2,n)
>>> torch.allclose(Q,Q2)
True

Single complex matrix

>>> n = 10
>>> theta = torch.rand(n**2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n,complex_case=True)
>>> theta2 = from_unitary_expskewh(Q,complex_case=True)
>>> torch.allclose(theta,theta2)
False
>>> Q2 = to_unitary_expskewh(theta2,n,complex_case=True)
>>> torch.allclose(Q,Q2)
True

Two matrices

>>> n = 10
>>> theta = torch.rand(2,n*(n-1)//2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n)
>>> theta2 = from_unitary_expskewh(Q)
>>> torch.allclose(theta,theta2)
False
>>> Q2 = to_unitary_expskewh(theta2,n)
>>> torch.allclose(Q,Q2)
True

Two complex matrices

>>> n = 3
>>> theta = torch.rand(2,n**2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n,complex_case=True)
>>> theta2 = from_unitary_expskewh(Q,complex_case=True)
>>> Q2 = to_unitary_expskewh(theta2,n,complex_case=True)
>>> torch.allclose(Q,Q2)
True

Batch support

>>> n = 10
>>> theta = torch.rand(2,3,4,n*(n-1)//2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n)
>>> theta2 = from_unitary_expskewh(Q)
>>> Q2 = to_unitary_expskewh(theta2,n)
>>> torch.allclose(Q,Q2)
True
>>> theta = torch.rand(2,3,4,n**2,generator=rng)
>>> Q = to_unitary_expskewh(theta,n,complex_case=True)
>>> theta2 = from_unitary_expskewh(Q,complex_case=True)
>>> Q2 = to_unitary_expskewh(theta2,n,complex_case=True)
>>> torch.allclose(Q,Q2)
True
Source code in agsutil/utils.py 
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def from_unitary_expskewh(Q, complex_case=False):
    r"""
    Transform from a unitary matrix using the exponential of a skew Hermitian matrix.

    Args:
        Q (torch.Tensor): With shape `(*theta.shape[:-1],n,n)` unitary matrices
        complex_case (bool): If `True`, parameterize a complex matrix, otherwise a real one. 

    Returns:
        theta (torch.Tensor): With `theta.size(-1) == n*(n-1)//2` in the real case and `theta.size(-1) == n**2` in the complex case.  
            Note that this `theta` is not unique, as shown in the following doctests.

    Examples:
        >>> torch.set_default_dtype(torch.float64)
        >>> rng = torch.Generator().manual_seed(7)

    Single matrix

        >>> n = 10
        >>> theta = torch.rand(n*(n-1)//2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n)
        >>> theta2 = from_unitary_expskewh(Q)
        >>> torch.allclose(theta,theta2)
        False
        >>> Q2 = to_unitary_expskewh(theta2,n)
        >>> torch.allclose(Q,Q2)
        True

    Single complex matrix

        >>> n = 10
        >>> theta = torch.rand(n**2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n,complex_case=True)
        >>> theta2 = from_unitary_expskewh(Q,complex_case=True)
        >>> torch.allclose(theta,theta2)
        False
        >>> Q2 = to_unitary_expskewh(theta2,n,complex_case=True)
        >>> torch.allclose(Q,Q2)
        True

    Two matrices

        >>> n = 10
        >>> theta = torch.rand(2,n*(n-1)//2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n)
        >>> theta2 = from_unitary_expskewh(Q)
        >>> torch.allclose(theta,theta2)
        False
        >>> Q2 = to_unitary_expskewh(theta2,n)
        >>> torch.allclose(Q,Q2)
        True

    Two complex matrices

        >>> n = 3
        >>> theta = torch.rand(2,n**2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n,complex_case=True)
        >>> theta2 = from_unitary_expskewh(Q,complex_case=True)
        >>> Q2 = to_unitary_expskewh(theta2,n,complex_case=True)
        >>> torch.allclose(Q,Q2)
        True

    Batch support

        >>> n = 10
        >>> theta = torch.rand(2,3,4,n*(n-1)//2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n)
        >>> theta2 = from_unitary_expskewh(Q)
        >>> Q2 = to_unitary_expskewh(theta2,n)
        >>> torch.allclose(Q,Q2)
        True
        >>> theta = torch.rand(2,3,4,n**2,generator=rng)
        >>> Q = to_unitary_expskewh(theta,n,complex_case=True)
        >>> theta2 = from_unitary_expskewh(Q,complex_case=True)
        >>> Q2 = to_unitary_expskewh(theta2,n,complex_case=True)
        >>> torch.allclose(Q,Q2)
        True
    """
    n = Q.size(-1)
    L = logm_unitary(Q)
    iut = torch.triu_indices(n,n,offset=1,device=Q.device)
    if complex_case:
        idiag = torch.arange(n,device=Q.device)
        diag = L[...,idiag,idiag].imag
        off_diag = L[...,iut[0],iut[1]]
        theta = torch.cat([diag,off_diag.real,off_diag.imag],dim=-1)
    else:
        theta = L[...,iut[0],iut[1]].real
    return theta

to_unitary_qr 

to_unitary_qr(A)

Transform to a unitary matrix using the QR decomposition.

Parameters:

Name Type Description Default
A Tensor

With A.shape == (...,n,n).

 required

Returns:

Name Type Description
Q Tensor

With shape (*A.shape[:-1],n,n) unitary matrices

Examples:

>>> torch.set_default_dtype(torch.float64)
>>> rng = torch.Generator().manual_seed(7)

Single matrix

>>> n = 5
>>> A = torch.rand(n,n,generator=rng)
>>> Q = to_unitary_qr(A)
>>> Q 
tensor([[ 0.1819,  0.2300,  0.8956,  0.3042,  0.1391],
        [ 0.5466, -0.1828, -0.3033,  0.7517, -0.1037],
        [ 0.0410,  0.0450,  0.1626, -0.0886, -0.9808],
        [ 0.5518, -0.6553,  0.2046, -0.4684,  0.0693],
        [ 0.6017,  0.6944, -0.1939, -0.3392,  0.0555]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
True

Single complex matrix

>>> n = 4
>>> A = torch.rand(n,n,generator=rng,dtype=torch.complex128)
>>> Q = to_unitary_qr(A)
>>> Q 
tensor([[ 0.3820+4.9592e-01j,  0.6227-3.3313e-01j,  0.2301+4.5375e-04j,
         -0.1762-1.5935e-01j],
        [ 0.3750+1.7952e-01j, -0.4719+1.6687e-01j,  0.1957+5.4894e-02j,
          0.2213-6.9735e-01j],
        [ 0.4702+1.8505e-01j,  0.0477+2.3842e-01j, -0.8138+2.9071e-02j,
          0.0823+1.2523e-01j],
        [ 0.4241+1.1515e-02j,  0.0064+4.3764e-01j,  0.4874-7.1170e-02j,
          0.2837+5.5258e-01j]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
True

Two matrices

>>> n = 3
>>> A = torch.rand(2,n,n,generator=rng)
>>> Q = to_unitary_qr(A)
>>> Q 
tensor([[[ 0.6587,  0.2967, -0.6914],
         [ 0.6269, -0.7246,  0.2864],
         [ 0.4160,  0.6221,  0.6633]],
<BLANKLINE>
        [[ 0.5190,  0.8009, -0.2985],
         [ 0.6364, -0.1289,  0.7605],
         [ 0.5706, -0.5847, -0.5766]]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
True

Two complex matrices

>>> n = 3
>>> A = torch.rand(2,n,n,generator=rng,dtype=torch.complex128)
>>> Q = to_unitary_qr(A)
>>> Q 
tensor([[[ 0.5834+0.1938j, -0.0560+0.3895j, -0.3185+0.6048j],
         [ 0.4323+0.4517j,  0.6040-0.4547j,  0.0490-0.1873j],
         [ 0.4700+0.1009j, -0.3015+0.4275j,  0.4046-0.5759j]],
<BLANKLINE>
        [[ 0.2769+0.2957j, -0.1093+0.8955j, -0.0100+0.1480j],
         [ 0.0858+0.6104j,  0.0647-0.1696j,  0.7218-0.2572j],
         [ 0.2022+0.6443j,  0.2591-0.2933j, -0.5677+0.2621j]]])
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
True

Batch support

>>> n = 10
>>> A = torch.rand(2,3,4,n,n,generator=rng)
>>> Q = to_unitary_qr(A)
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
True
>>> A = torch.rand(2,3,4,n,n,generator=rng,dtype=torch.complex128)
>>> Q = to_unitary_qr(A)
>>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
True
Source code in agsutil/utils.py 
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def to_unitary_qr(A):
    r"""
    Transform to a unitary matrix using the QR decomposition.

    Args:
        A (torch.Tensor): With `A.shape == (...,n,n)`.

    Returns:
        Q (torch.Tensor): With shape `(*A.shape[:-1],n,n)` unitary matrices

    Examples:
        >>> torch.set_default_dtype(torch.float64)
        >>> rng = torch.Generator().manual_seed(7)

    Single matrix

        >>> n = 5
        >>> A = torch.rand(n,n,generator=rng)
        >>> Q = to_unitary_qr(A)
        >>> Q 
        tensor([[ 0.1819,  0.2300,  0.8956,  0.3042,  0.1391],
                [ 0.5466, -0.1828, -0.3033,  0.7517, -0.1037],
                [ 0.0410,  0.0450,  0.1626, -0.0886, -0.9808],
                [ 0.5518, -0.6553,  0.2046, -0.4684,  0.0693],
                [ 0.6017,  0.6944, -0.1939, -0.3392,  0.0555]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
        True

    Single complex matrix

        >>> n = 4
        >>> A = torch.rand(n,n,generator=rng,dtype=torch.complex128)
        >>> Q = to_unitary_qr(A)
        >>> Q 
        tensor([[ 0.3820+4.9592e-01j,  0.6227-3.3313e-01j,  0.2301+4.5375e-04j,
                 -0.1762-1.5935e-01j],
                [ 0.3750+1.7952e-01j, -0.4719+1.6687e-01j,  0.1957+5.4894e-02j,
                  0.2213-6.9735e-01j],
                [ 0.4702+1.8505e-01j,  0.0477+2.3842e-01j, -0.8138+2.9071e-02j,
                  0.0823+1.2523e-01j],
                [ 0.4241+1.1515e-02j,  0.0064+4.3764e-01j,  0.4874-7.1170e-02j,
                  0.2837+5.5258e-01j]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
        True

    Two matrices

        >>> n = 3
        >>> A = torch.rand(2,n,n,generator=rng)
        >>> Q = to_unitary_qr(A)
        >>> Q 
        tensor([[[ 0.6587,  0.2967, -0.6914],
                 [ 0.6269, -0.7246,  0.2864],
                 [ 0.4160,  0.6221,  0.6633]],
        <BLANKLINE>
                [[ 0.5190,  0.8009, -0.2985],
                 [ 0.6364, -0.1289,  0.7605],
                 [ 0.5706, -0.5847, -0.5766]]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
        True

    Two complex matrices

        >>> n = 3
        >>> A = torch.rand(2,n,n,generator=rng,dtype=torch.complex128)
        >>> Q = to_unitary_qr(A)
        >>> Q 
        tensor([[[ 0.5834+0.1938j, -0.0560+0.3895j, -0.3185+0.6048j],
                 [ 0.4323+0.4517j,  0.6040-0.4547j,  0.0490-0.1873j],
                 [ 0.4700+0.1009j, -0.3015+0.4275j,  0.4046-0.5759j]],
        <BLANKLINE>
                [[ 0.2769+0.2957j, -0.1093+0.8955j, -0.0100+0.1480j],
                 [ 0.0858+0.6104j,  0.0647-0.1696j,  0.7218-0.2572j],
                 [ 0.2022+0.6443j,  0.2591-0.2933j, -0.5677+0.2621j]]])
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
        True

    Batch support

        >>> n = 10
        >>> A = torch.rand(2,3,4,n,n,generator=rng)
        >>> Q = to_unitary_qr(A)
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q),torch.eye(n))
        True
        >>> A = torch.rand(2,3,4,n,n,generator=rng,dtype=torch.complex128)
        >>> Q = to_unitary_qr(A)
        >>> torch.allclose(torch.einsum("...ji,...jk->...ik",Q,Q.conj()),torch.eye(n,dtype=torch.complex128))
        True
    """
    n = A.size(-1)
    assert A.shape[-2:]==(n,n)
    Q,R = torch.linalg.qr(A)
    phases = torch.sgn(torch.diagonal(R,dim1=-2,dim2=-1))
    return Q*phases.unsqueeze(-2)

get_torch_rng 

get_torch_rng(seed=None, device=None)

Get a torch.Generator() a random seed when seed=None or a fixed seeed when seed is an int.
This is necessary because torch.Generator() uses a fixed default seed.

Parameters:

Name Type Description Default
seed Union[None, int]

Random seed. If None.

None
device device

Device on which to place the generator.

 None

Returns:

Name Type Description
rng Generator

The random number generator
Source code in agsutil/utils.py 
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def get_torch_rng(seed=None, device=None):
    r"""
    Get a `torch.Generator()` a random seed when `seed=None` or a fixed seeed when `seed` is an int.  
    This is necessary because torch.Generator() uses a fixed default seed.  


    Args:
        seed (Union[None,int]): Random seed. If None. 
        device (torch.device): Device on which to place the generator.

    Returns: 
        rng (torch.Generator): The random number generator
    """
    if device is None: 
        device = torch.get_default_device()
    if seed is None: 
        rng = torch.Generator(device=device)
        rng.seed()
    else:
        rng = torch.Generator(device=device).manual_seed(seed)
    return rng

logmultinomialcoeff 

logmultinomialcoeff(n, *ks)

\(\log \binom{n}{k_1,\dots,k_m} = \log \left(\frac{n!}{k_1! \cdots k_m!}\right)\).
Note that we do not enforce \(k_1+\cdots+k_m=n\).

Parameters:

Name Type Description Default
n Tensor

\(n\).

 required
*ks Tuple

\(k_1,\dots,k_m\) where \(k_j\) is a torch.Tensor.

 ()

Returns:

Name Type Description
y Tensor

\(\log \binom{n}{k_1,\dots,k_m}\).

Examples:

>>> n = torch.arange(6,12)
>>> k1 = torch.arange(0,2)
>>> k2 = torch.arange(2,4)
>>> k3 = torch.arange(4,6)
>>> logmultinomialcoeff(n[None,:],k1[:,None],k2[:,None],k3[:,None])
tensor([[2.7081e+00, 4.6540e+00, 6.7334e+00, 8.9306e+00, 1.1233e+01, 1.3631e+01],
        [8.8818e-16, 1.9459e+00, 4.0254e+00, 6.2226e+00, 8.5252e+00, 1.0923e+01]])
>>> logmultinomialcoeff(n[:,None,None,None],k1[None,:,None,None],k2[None,None,:,None],k3[None,None,None,:]).shape
torch.Size([6, 2, 2, 2])
Source code in agsutil/utils.py 
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def logmultinomialcoeff(n, *ks):
    r"""
    $\log \binom{n}{k_1,\dots,k_m} = \log \left(\frac{n!}{k_1! \cdots k_m!}\right)$.  
    Note that we do not enforce $k_1+\cdots+k_m=n$.

    Args:
        n (torch.Tensor): $n$.
        *ks (Tuple): $k_1,\dots,k_m$ where $k_j$ is a `torch.Tensor`.

    Returns:
        y (torch.Tensor): $\log \binom{n}{k_1,\dots,k_m}$.

    Examples:
        >>> n = torch.arange(6,12)
        >>> k1 = torch.arange(0,2)
        >>> k2 = torch.arange(2,4)
        >>> k3 = torch.arange(4,6)
        >>> logmultinomialcoeff(n[None,:],k1[:,None],k2[:,None],k3[:,None])
        tensor([[2.7081e+00, 4.6540e+00, 6.7334e+00, 8.9306e+00, 1.1233e+01, 1.3631e+01],
                [8.8818e-16, 1.9459e+00, 4.0254e+00, 6.2226e+00, 8.5252e+00, 1.0923e+01]])
        >>> logmultinomialcoeff(n[:,None,None,None],k1[None,:,None,None],k2[None,None,:,None],k3[None,None,None,:]).shape
        torch.Size([6, 2, 2, 2])
    """ 
    assert (n>=0).all()
    m = len(ks) 
    assert ((k>=0).all() for k in ks)
    y = torch.lgamma(n+1)
    for k in ks:
        y = y-torch.lgamma(k+1)
    return y

multinomialcoeff 

multinomialcoeff(n, *ks)

\(\binom{n}{k_1,\dots,k_m} = \frac{n!}{k_1! \cdots k_m!}\).
Note that we do not enforce \(k_1+\cdots+k_m=n\), so we do not round the result to the nearest integer.

Parameters:

Name Type Description Default
n Tensor

\(n\).

 required
*ks Tuple

\(k_1,\dots,k_m\) where \(k_j\) is a torch.Tensor.

 ()

Returns:

Name Type Description
y Tensor

\(\binom{n}{k_1,\dots,k_m}\).

Examples:

>>> n = torch.arange(6,12)
>>> k1 = torch.arange(0,2)
>>> k2 = torch.arange(2,4)
>>> k3 = torch.arange(4,6)
>>> multinomialcoeff(n[None,:],k1[:,None],k2[:,None],k3[:,None])
tensor([[1.5000e+01, 1.0500e+02, 8.4000e+02, 7.5600e+03, 7.5600e+04, 8.3160e+05],
        [1.0000e+00, 7.0000e+00, 5.6000e+01, 5.0400e+02, 5.0400e+03, 5.5440e+04]])
>>> multinomialcoeff(n[:,None,None,None],k1[None,:,None,None],k2[None,None,:,None],k3[None,None,None,:]).shape
torch.Size([6, 2, 2, 2])
Source code in agsutil/utils.py 
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def multinomialcoeff(n, *ks):
    r"""
    $\binom{n}{k_1,\dots,k_m} = \frac{n!}{k_1! \cdots k_m!}$.  
    Note that we do not enforce $k_1+\cdots+k_m=n$, so we do not round the result to the nearest integer.

    Args:
        n (torch.Tensor): $n$.
        *ks (Tuple): $k_1,\dots,k_m$ where $k_j$ is a `torch.Tensor`.

    Returns:
        y (torch.Tensor): $\binom{n}{k_1,\dots,k_m}$.

    Examples:
        >>> n = torch.arange(6,12)
        >>> k1 = torch.arange(0,2)
        >>> k2 = torch.arange(2,4)
        >>> k3 = torch.arange(4,6)
        >>> multinomialcoeff(n[None,:],k1[:,None],k2[:,None],k3[:,None])
        tensor([[1.5000e+01, 1.0500e+02, 8.4000e+02, 7.5600e+03, 7.5600e+04, 8.3160e+05],
                [1.0000e+00, 7.0000e+00, 5.6000e+01, 5.0400e+02, 5.0400e+03, 5.5440e+04]])
        >>> multinomialcoeff(n[:,None,None,None],k1[None,:,None,None],k2[None,None,:,None],k3[None,None,None,:]).shape
        torch.Size([6, 2, 2, 2])
    """ 
    return torch.exp(logmultinomialcoeff(n,*ks))

logfactorial 

logfactorial(n)

\(\log(n!)\)

Parameters:

Name Type Description Default
n Tensor

\(n\).

 required

Returns:

Name Type Description
y Tensor

\(\log(n!)\).

Examples:

>>> logfactorial(torch.arange(1,8))
tensor([0.0000, 0.6931, 1.7918, 3.1781, 4.7875, 6.5793, 8.5252])
Source code in agsutil/utils.py 
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def logfactorial(n):
    r"""
    $\log(n!)$

    Args:
        n (torch.Tensor): $n$.

    Returns:
        y (torch.Tensor): $\log(n!)$.

    Examples:
        >>> logfactorial(torch.arange(1,8))
        tensor([0.0000, 0.6931, 1.7918, 3.1781, 4.7875, 6.5793, 8.5252])
    """ 
    return logmultinomialcoeff(n)

factorial 

factorial(n)

\(n!\)

Parameters:

Name Type Description Default
n Tensor

\(n\).

 required

Returns:

Name Type Description
y Tensor

\(n!\).

Examples:

>>> factorial(torch.arange(1,8))
tensor([   1,    2,    6,   24,  120,  720, 5040])
Source code in agsutil/utils.py 
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def factorial(n):
    r"""
    $n!$

    Args:
        n (torch.Tensor): $n$.

    Returns:
        y (torch.Tensor): $n!$.

    Examples:
        >>> factorial(torch.arange(1,8))
        tensor([   1,    2,    6,   24,  120,  720, 5040])
    """ 
    return multinomialcoeff(n).round().to(int)

logcomb 

logcomb(n, k)

\(\log \binom{n}{k} = \log \left(\frac{n!}{k!(n-k)!}\right)\)

Parameters:

Name Type Description Default
n Tensor

\(n\).

 required
k Tensor

\(k\).

 required

Returns:

Name Type Description
y Tensor

\(\log \binom{n}{k}\).

Examples:

>>> logcomb(torch.arange(8)[:,None],torch.arange(6)[None,:])
tensor([[0.0000,   -inf,   -inf,   -inf,   -inf,   -inf],
        [0.0000, 0.0000,   -inf,   -inf,   -inf,   -inf],
        [0.0000, 0.6931, 0.0000,   -inf,   -inf,   -inf],
        [0.0000, 1.0986, 1.0986, 0.0000,   -inf,   -inf],
        [0.0000, 1.3863, 1.7918, 1.3863, 0.0000,   -inf],
        [0.0000, 1.6094, 2.3026, 2.3026, 1.6094, 0.0000],
        [0.0000, 1.7918, 2.7081, 2.9957, 2.7081, 1.7918],
        [0.0000, 1.9459, 3.0445, 3.5553, 3.5553, 3.0445]])
Source code in agsutil/utils.py 
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def logcomb(n,k):
    r"""
    $\log \binom{n}{k} = \log \left(\frac{n!}{k!(n-k)!}\right)$

    Args:
        n (torch.Tensor): $n$.
        k (torch.Tensor): $k$.

    Returns:
        y (torch.Tensor): $\log \binom{n}{k}$.

    Examples:
        >>> logcomb(torch.arange(8)[:,None],torch.arange(6)[None,:])
        tensor([[0.0000,   -inf,   -inf,   -inf,   -inf,   -inf],
                [0.0000, 0.0000,   -inf,   -inf,   -inf,   -inf],
                [0.0000, 0.6931, 0.0000,   -inf,   -inf,   -inf],
                [0.0000, 1.0986, 1.0986, 0.0000,   -inf,   -inf],
                [0.0000, 1.3863, 1.7918, 1.3863, 0.0000,   -inf],
                [0.0000, 1.6094, 2.3026, 2.3026, 1.6094, 0.0000],
                [0.0000, 1.7918, 2.7081, 2.9957, 2.7081, 1.7918],
                [0.0000, 1.9459, 3.0445, 3.5553, 3.5553, 3.0445]])
    """ 
    return logmultinomialcoeff(n,k,n-k)

comb 

comb(n, k)

\(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)

Parameters:

Name Type Description Default
n Tensor

\(n\).

 required
k Tensor

\(k\).

 required

Returns:

Name Type Description
y Tensor

\(\binom{n}{k}\).

Examples:

>>> comb(torch.arange(8)[:,None],torch.arange(6)[None,:])
tensor([[ 1,  0,  0,  0,  0,  0],
        [ 1,  1,  0,  0,  0,  0],
        [ 1,  2,  1,  0,  0,  0],
        [ 1,  3,  3,  1,  0,  0],
        [ 1,  4,  6,  4,  1,  0],
        [ 1,  5, 10, 10,  5,  1],
        [ 1,  6, 15, 20, 15,  6],
        [ 1,  7, 21, 35, 35, 21]])
Source code in agsutil/utils.py 
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def comb(n,k):
    r"""
    $\binom{n}{k} = \frac{n!}{k!(n-k)!}$

    Args:
        n (torch.Tensor): $n$.
        k (torch.Tensor): $k$.

    Returns:
        y (torch.Tensor): $\binom{n}{k}$.

    Examples:
        >>> comb(torch.arange(8)[:,None],torch.arange(6)[None,:])
        tensor([[ 1,  0,  0,  0,  0,  0],
                [ 1,  1,  0,  0,  0,  0],
                [ 1,  2,  1,  0,  0,  0],
                [ 1,  3,  3,  1,  0,  0],
                [ 1,  4,  6,  4,  1,  0],
                [ 1,  5, 10, 10,  5,  1],
                [ 1,  6, 15, 20, 15,  6],
                [ 1,  7, 21, 35, 35, 21]])
    """ 
    return multinomialcoeff(n,k,n-k).round().to(int)

enumerate_sums 

enumerate_sums(s, t)

Generator of all possible ways to choose \(s \in \mathbb{N}_0\) non-negative integers which sum to \(t \in \mathbb{N}_0\).
There are a total of \(\binom{t+s-1}{s-1} = \frac{(t+s-1)!}{t!(s-1)!}\) choices.

Parameters:

Name Type Description Default
s int

Number of integers to choose.

required
t int

Number of items to allocate.

required

Returns:

Name Type Description
g generator

Genrator of tuples of length \(s\).

Examples:

2 non-negative integers that sum to 3 

>>> for v in enumerate_sums(2,3): 
...     print(v)
(0, 3)
(1, 2)
(2, 1)
(3, 0)
>>> len_enumerate_sums(2,3)
4

3 non-negative integers that sum to 2 

>>> for v in enumerate_sums(3,2): 
...     print(v)
(0, 0, 2)
(0, 1, 1)
(0, 2, 0)
(1, 0, 1)
(1, 1, 0)
(2, 0, 0)
>>> len_enumerate_sums(3,2)
6
Source code in agsutil/utils.py 
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def enumerate_sums(s, t):
    r"""
    Generator of all possible ways to choose $s \in \mathbb{N}_0$ non-negative integers which sum to $t \in \mathbb{N}_0$.  
    There are a total of $\binom{t+s-1}{s-1} = \frac{(t+s-1)!}{t!(s-1)!}$ choices. 

    Args:
        s (int): Number of integers to choose. 
        t (int): Number of items to allocate. 

    Returns: 
        g (generator): Genrator of tuples of length $s$. 

    Examples: 

    2 non-negative integers that sum to 3 

        >>> for v in enumerate_sums(2,3): 
        ...     print(v)
        (0, 3)
        (1, 2)
        (2, 1)
        (3, 0)
        >>> len_enumerate_sums(2,3)
        4

    3 non-negative integers that sum to 2 

        >>> for v in enumerate_sums(3,2): 
        ...     print(v)
        (0, 0, 2)
        (0, 1, 1)
        (0, 2, 0)
        (1, 0, 1)
        (1, 1, 0)
        (2, 0, 0)
        >>> len_enumerate_sums(3,2)
        6
    """ 
    for combo in itertools.combinations(range(t+s-1),s-1):
        bars = (-1,)+combo+(t+s-1,)
        yield tuple(bars[i+1]-bars[i]-1 for i in range(s))

len_enumerate_sums 

len_enumerate_sums(s, t)

\(\binom{t+s-1}{s-1} = \frac{(t+s-1)!}{t!(s-1)!}\), the number of ways to choose \(s \in \mathbb{N}_0\) non-negative integers which sum to \(t \in \mathbb{N}_0\).

Parameters:

Name Type Description Default
s int

Number of integers to choose.

required
t int

Number of items to allocate.

required

Returns:

Name Type Description
y int

\(\binom{t+s-1}{s-1}\).

Examples:

2 non-negative integers that sum to 3 

>>> len_enumerate_sums(2,3)
4

3 non-negative integers that sum to 2 

>>> len_enumerate_sums(3,2)
6
Source code in agsutil/utils.py 
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def len_enumerate_sums(s, t):
    r"""
    $\binom{t+s-1}{s-1} = \frac{(t+s-1)!}{t!(s-1)!}$, the number of ways to choose $s \in \mathbb{N}_0$ non-negative integers which sum to $t \in \mathbb{N}_0$.  

    Args:
        s (int): Number of integers to choose. 
        t (int): Number of items to allocate. 

    Returns: 
        y (int): $\binom{t+s-1}{s-1}$. 

    Examples: 

    2 non-negative integers that sum to 3 

        >>> len_enumerate_sums(2,3)
        4

    3 non-negative integers that sum to 2 

        >>> len_enumerate_sums(3,2)
        6
    """ 
    return comb(torch.tensor(t+s-1),torch.tensor(s-1)).item()

enumerate_partitions 

enumerate_partitions(n, max_val=None)

Enumerate all partitions sizes of \(n\) objects.

Parameters:

Name Type Description Default
n int

Number of objects

 required

Returns:

Name Type Description
g generator

Generator of partitions.

Examples:

>>> for p in enumerate_partitions(3):
...     print(p)
[3]
[2, 1]
[1, 1, 1]
>>> for p in enumerate_partitions(5):
...     print(p)
[5]
[4, 1]
[3, 2]
[3, 1, 1]
[2, 2, 1]
[2, 1, 1, 1]
[1, 1, 1, 1, 1]
Source code in agsutil/utils.py 
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def enumerate_partitions(n, max_val=None):
    r"""
    Enumerate all partitions sizes of $n$ objects.

    Args:
        n (int): Number of objects

    Returns: 
        g (generator): Generator of partitions. 

    Examples: 
        >>> for p in enumerate_partitions(3):
        ...     print(p)
        [3]
        [2, 1]
        [1, 1, 1]
        >>> for p in enumerate_partitions(5):
        ...     print(p)
        [5]
        [4, 1]
        [3, 2]
        [3, 1, 1]
        [2, 2, 1]
        [2, 1, 1, 1]
        [1, 1, 1, 1, 1]
    """
    if max_val is None:
        max_val = n
    if n == 0:
        yield []
        return
    for i in range(min(n,max_val),0,-1):
        for p in enumerate_partitions(n-i,i):
            yield [i]+p

icdf_std_normal 

icdf_std_normal(x)

Inverse CDF of the standard normal distribution \(\mathcal{N}(0,1)\).

Examples:

>>> torch.set_default_dtype(torch.float64)
>>> rng = torch.Generator().manual_seed(7)

>>> x = torch.rand(3,4,generator=rng)
>>> g = icdf_std_normal(x)
>>> g 
tensor([[-0.5847, -0.6017,  1.0898,  0.4034],
        [ 1.4223,  0.9926, -0.5397,  0.1528],
        [ 0.7161,  0.7042, -1.5299, -1.5756]])
>>> g2 = icdf_normal = torch.distributions.Normal(loc=torch.zeros(1),scale=torch.ones(1)).icdf(x)
>>> torch.allclose(g,g2)
True
Source code in agsutil/utils.py 
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def icdf_std_normal(x):
    r"""
    Inverse CDF of the standard normal distribution $\mathcal{N}(0,1)$. 

    Examples:
        >>> torch.set_default_dtype(torch.float64)
        >>> rng = torch.Generator().manual_seed(7)

        >>> x = torch.rand(3,4,generator=rng)
        >>> g = icdf_std_normal(x)
        >>> g 
        tensor([[-0.5847, -0.6017,  1.0898,  0.4034],
                [ 1.4223,  0.9926, -0.5397,  0.1528],
                [ 0.7161,  0.7042, -1.5299, -1.5756]])
        >>> g2 = icdf_normal = torch.distributions.Normal(loc=torch.zeros(1),scale=torch.ones(1)).icdf(x)
        >>> torch.allclose(g,g2)
        True
    """
    return np.sqrt(2)*torch.erfinv(2*x-1)

matplotlib plotting utils

mpl_setup 

mpl_setup()

Setup matplotlib default parameters

Returns:

Name Type Description
mplparams dict

matplotlib helpful parameters
Source code in agsutil/plots.py 
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def mpl_setup():
    r""" 
    Setup matplotlib default parameters

    Returns:
        mplparams (dict): matplotlib helpful parameters"""
    from matplotlib import pyplot 
    import seaborn as sns
    pyplot.style.use("seaborn-v0_8-whitegrid")
    sns.set_palette("colorblind") # Options: "deep", "muted", "pastel", "bright", "dark", "colorblind
    COLORS = pyplot.rcParams['axes.prop_cycle'].by_key()['color']
    PW = 30 # page width in inches
    FS = 30 # font size
    LINESTYLES = [
        'solid',
        'dotted',
        'dashdot',
        'dashed',
        (0, (3, 5, 1, 5, 1, 5)),
        (5, (10, 3)),
        (0, (1, 1)),
        (0, (1, 10)),
        (0, (1, 5)),
        (0, (5, 5)),
        ]
    MARKERS = [
        "o",
        "s",
        "D",
        "P",
        "^",
        "v",
        "<",
        ">",
        "$a$",
        "$b$",
        "$c$",
        "$d$",
        ]
    pyplot.rcParams['xtick.labelsize'] = FS
    pyplot.rcParams['ytick.labelsize'] = FS
    pyplot.rcParams['ytick.labelsize'] = FS
    pyplot.rcParams['axes.titlesize'] = FS
    pyplot.rcParams['figure.titlesize'] = FS
    pyplot.rcParams["axes.labelsize"] = FS
    pyplot.rcParams['legend.fontsize'] = FS
    pyplot.rcParams['font.size'] = FS
    pyplot.rcParams['lines.linewidth'] = 5
    pyplot.rcParams['lines.markersize'] = 15
    mplparams = {
        "PW": PW,
        "FS": FS,
        "COLORS": COLORS,
        "LINESTYLES": LINESTYLES,
        "MARKERS": MARKERS,
        }
    return mplparams

set_aspect 

set_aspect(ax, ratio=1)

Set aspect ratio of the ax.

Parameters:

Name Type Description Default
ax Axes

axes to set the aspect ratio of.

required
ratio float

positive aspect ratio for the axis

 1
Source code in agsutil/plots.py 
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def set_aspect(ax, ratio=1):
    r"""
    Set aspect ratio of the ax. 

    Args:
        ax (Axes): axes to set the aspect ratio of. 
        ratio (float): positive aspect ratio for the axis
    """
    assert ratio>0
    xmin,xmax = ax.get_xlim()
    ymin,ymax = ax.get_ylim()
    aspect = ratio*(xmax-xmin)/(ymax-ymin)
    ax.set_aspect(aspect)