spd_learn.functional.cholesky_log#

class spd_learn.functional.cholesky_log(*args, **kwargs)[source]#

Bases: Function

Matrix logarithm via Cholesky decomposition (Log-Cholesky map).

This function computes the Log-Cholesky representation of an SPD matrix. Given an SPD matrix \(X\), it first computes the Cholesky decomposition:

\[X = LL^T\]

where \(L\) is a lower triangular matrix with positive diagonal entries. The Log-Cholesky map then applies the logarithm to the diagonal:

\[\log_{\text{chol}}(L) = \text{tril}(L, -1) + \text{diag}(\log(\text{diag}(L)))\]

This maps the SPD manifold to the vector space of lower triangular matrices with arbitrary diagonal entries.

Parameters:: X (torch.Tensor) – SPD matrix of shape (…, n, n).
Returns:: Log-Cholesky representation of X as a lower triangular matrix of shape (…, n, n).
Return type:: torch.Tensor

Notes

The backward pass is derived analytically [Lin, 2019]. For the forward map \(f: X \mapsto \log_{\text{chol}}(L)\) where \(X = LL^T\), the gradient computation involves the Cholesky derivative.

See also

cholesky_exp: Inverse mapping from Log-Cholesky space to SPD.
log_cholesky_distance(): Distance under Log-Cholesky metric.
matrix_log(): Matrix logarithm via eigendecomposition.

static backward(ctx, grad_output)[source]#

Define a formula for differentiating the operation with backward mode automatic differentiation.

This function is to be overridden by all subclasses. (Defining this function is equivalent to defining the vjp function.)

It must accept a context ctx as the first argument, followed by as many outputs as the forward() returned (None will be passed in for non tensor outputs of the forward function), and it should return as many tensors, as there were inputs to forward(). Each argument is the gradient w.r.t the given output, and each returned value should be the gradient w.r.t. the corresponding input. If an input is not a Tensor or is a Tensor not requiring grads, you can just pass None as a gradient for that input.

The context can be used to retrieve tensors saved during the forward pass. It also has an attribute ctx.needs_input_grad as a tuple of booleans representing whether each input needs gradient. E.g., backward() will have ctx.needs_input_grad[0] = True if the first input to forward() needs gradient computed w.r.t. the output.

static forward(ctx, X)[source]#

Define the forward of the custom autograd Function.

This function is to be overridden by all subclasses. There are two ways to define forward:

Usage 1 (Combined forward and ctx):

@staticmethod
def forward(ctx: Any, *args: Any, **kwargs: Any) -> Any:
    pass

It must accept a context ctx as the first argument, followed by any number of arguments (tensors or other types).
See Combined or separate forward() and setup_context() for more details

Usage 2 (Separate forward and ctx):

@staticmethod
def forward(*args: Any, **kwargs: Any) -> Any:
    pass

@staticmethod
def setup_context(ctx: Any, inputs: Tuple[Any, ...], output: Any) -> None:
    pass

The forward no longer accepts a ctx argument.
Instead, you must also override the torch.autograd.Function.setup_context() staticmethod to handle setting up the ctx object. output is the output of the forward, inputs are a Tuple of inputs to the forward.
See Extending torch.autograd for more details

The context can be used to store arbitrary data that can be then retrieved during the backward pass. Tensors should not be stored directly on ctx (though this is not currently enforced for backward compatibility). Instead, tensors should be saved either with ctx.save_for_backward() if they are intended to be used in backward (equivalently, vjp) or ctx.save_for_forward() if they are intended to be used for in jvp.