spd_learn.modules.ExpEig#

class spd_learn.modules.ExpEig(upper=False, flatten=False, autograd=False, device=None, dtype=None)[source]#

Bases: Module

Exponential Eigenvalue Layer (ExpEig).

This layer maps symmetric matrices from the tangent space back to the SPD manifold via the matrix exponential [Huang and Van Gool, 2017]. It is the inverse operation of LogEig.

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

where \(X = U \Lambda U^\top\) is the eigendecomposition of the symmetric input matrix, and \(\exp(\Lambda)\) applies the exponential element-wise to eigenvalues.

Geometric Interpretation

The exponential map \(\exp_I: T_I\mathcal{S}_{++}^n \to \mathcal{S}_{++}^n\) projects tangent vectors at the identity to points on the SPD manifold. This enables:

  1. Manifold reconstruction: After processing in tangent space (e.g., via Euclidean layers), data can be projected back to valid SPD matrices.

  2. Generative models: Sampling in tangent space and mapping to manifold.

  3. Residual connections: Combined with LogEig for manifold-aware skip connections.

Since \(\exp(\cdot)\) always produces positive values, the output is guaranteed to be SPD regardless of the input symmetric matrix.

Parameters:

  • upper (bool, default=False) – If True, assumes the input is an upper-triangular vectorized symmetric matrix of shape (..., n(n+1)/2) and reconstructs the full symmetric matrix before applying the exponential.

  • flatten (bool, default=False) – If True, assumes the input is a flattened symmetric matrix of shape (..., n*n) and reshapes it to (..., n, n) before applying the exponential.

  • autograd (bool, default=False) – Whether to use the autograd backend.

See also

LogEig

Inverse operation, maps SPD to tangent space.

ReEig

Eigenvalue rectification for numerical stability.

matrix_exp()

Functional version.

LogEuclideanResidual

Uses exp/log for manifold residuals.

autograd_: Tensor#
forward(X: Tensor) → Tensor[source]#

Forward pass of the ExpEig layer.

Parameters:

X (torch.Tensor) – Input symmetric matrix or vectorized tangent vector. If upper=True, expects upper-triangular vectorized input of shape (..., n(n+1)/2). If flatten=True, expects flattened input of shape (..., n*n). Otherwise, expects a full symmetric matrix (..., n, n).

Returns:

The output SPD matrix of shape (..., n, n).

Return type:

torch.Tensor