spd_learn.functional.cholesky_exp#

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

Bases: Function

Inverse of the Log-Cholesky map (Cholesky exponential).

This function reconstructs an SPD matrix from its Log-Cholesky representation. Given a lower triangular matrix \(Y\) (with arbitrary diagonal entries), it computes:

\[L = \text{tril}(Y, -1) + \text{diag}(\exp(\text{diag}(Y)))\]

\[X = LL^T\]

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

Notes

This is the inverse of cholesky_log(). The composition cholesky_exp(cholesky_log(X)) = X for any SPD matrix X.

See also

cholesky_log: Forward mapping from SPD to Log-Cholesky space.
log_cholesky_mean(): Fréchet mean under Log-Cholesky metric.
matrix_exp(): Matrix exponential 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, Y)[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.