spd_learn.functional.exp_map_lem#

spd_learn.functional.exp_map_lem(P, V)[source]#

Riemannian exponential map under the Log-Euclidean metric.

Maps a tangent vector \(V\) at base point \(P\) to a point on the SPD manifold by shooting along the geodesic in direction \(V\).

Under the Log-Euclidean metric, the exponential map is:

\[\text{Exp}_P(V) = \exp(\log(P) + V)\]

where \(V\) is a symmetric matrix representing a tangent vector at \(P\).

Parameters:

P (torch.Tensor) – Base point on the SPD manifold with shape (…, n, n).
V (torch.Tensor) – Tangent vector at P (symmetric matrix) with shape (…, n, n).

Returns:

Point on the SPD manifold with shape (…, n, n).

Return type:

torch.Tensor

Notes

Under the Log-Euclidean metric, the SPD manifold is globally flat (zero curvature), so the exponential map is a global diffeomorphism. The tangent space at any point can be identified with the space of symmetric matrices.