spd_learn.functional.matrix_exp#

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

Bases: Function

Matrix exponential of a symmetric matrix.

This function computes the matrix exponential of a symmetric matrix. For each symmetric matrix \(X\), the matrix exponential is computed via eigenvalue decomposition:

\[X = U \Lambda U^\top\]

\[\exp(X) = U \exp(\Lambda) U^\top\]

where \(\Lambda\) is the diagonal matrix of eigenvalues and \(U\) is the matrix of eigenvectors. The exponential is applied element-wise to the eigenvalues.

This function is used to map matrices back to the SPD manifold after operations have been performed in the vector space using the matrix logarithm.

Parameters:: X (torch.Tensor) – Symmetric matrix of shape (…, n, n).
Returns:: Matrix exponential of X.
Return type:: torch.Tensor

See also

matrix_log: Inverse operation, maps SPD to tangent space.
log_euclidean_mean(): Mean under Log-Euclidean metric uses exp/log.
ExpEig: Neural network layer using matrix exponential.
cholesky_exp: Alternative via Cholesky decomposition.

static applied_fct(s)[source]#

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 derivative(s)[source]#

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.