spd_learn.functional.log_map_lem#
- spd_learn.functional.log_map_lem(P, Q)[source]#
Riemannian logarithmic map under the Log-Euclidean metric.
Maps a point \(Q\) on the SPD manifold to a tangent vector at base point \(P\). This is the inverse of the exponential map.
Under the Log-Euclidean metric, the logarithmic map is:
\[\text{Log}_P(Q) = \log(Q) - \log(P)\]The result is a symmetric matrix representing the tangent vector at \(P\) that points towards \(Q\).
- Parameters:
P (torch.Tensor) – Base point on the SPD manifold with shape (…, n, n).
Q (torch.Tensor) – Target point on the SPD manifold with shape (…, n, n).
- Returns:
Tangent vector at P (symmetric matrix) with shape (…, n, n).
- Return type:
Notes
The norm of the tangent vector equals the Log-Euclidean distance:
\[\|\text{Log}_P(Q)\|_F = d_{\text{LEM}}(P, Q)\]See also
exp_map_lem()Inverse operation (exponential map).
log_euclidean_distance()Distance under Log-Euclidean metric.
parallel_transport_lem()Parallel transport under LEM.
References
See [Arsigny et al., 2007] for more details.