spd_learn.functional.log_euclidean_distance#

spd_learn.functional.log_euclidean_distance(A, B)[source]#

Computes the Log-Euclidean distance between SPD matrices.

The Log-Euclidean Metric (LEM) simplifies computations by using the matrix logarithm to map the SPD manifold diffeomorphically to the Euclidean vector space of symmetric matrices. The mapping \(\log: \mathcal{S}_{++}^n \to \mathcal{S}^n\) is a global diffeomorphism.

The Log-Euclidean distance is defined as the Frobenius norm of the difference of the matrix logarithms:

\[d_{\text{LEM}}(A, B) = \| \log(A) - \log(B) \|_F\]

This metric endows SPD matrices with a commutative Lie group structure, where the group operation is \(A \odot B = \exp(\log(A) + \log(B))\).

Parameters:

A (torch.Tensor) – SPD matrices with shape (…, n, n).
B (torch.Tensor) – SPD matrices with shape (…, n, n). Must be broadcastable with A.

Returns:

Distances with shape (…).

Return type:

torch.Tensor

Notes

Unlike the Affine-Invariant Riemannian Metric (AIRM), the Log-Euclidean metric is not affine-invariant. It is invariant only under orthogonal transformations (rotations): \(d(QAQ^\top, QBQ^\top) = d(A, B)\) for orthogonal \(Q\). However, it offers computational advantages as all operations reduce to standard Euclidean operations in the log-domain.