spd_learn.functional.log_cholesky_distance#

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

Compute the distance in the Log-Cholesky metric.

The Log-Cholesky distance between two SPD matrices \(A\) and \(B\) is the Frobenius norm of the difference of their Log-Cholesky representations:

\[d_{LC}(A, B) = \|\log_{\text{chol}}(L_A) - \log_{\text{chol}}(L_B)\|_F\]

where \(A = L_A L_A^T\) and \(B = L_B L_B^T\) are the Cholesky decompositions.

Parameters:

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

Returns:

Distances of shape (…) (batch dimensions).

Return type:

torch.Tensor

Notes

This metric is computationally cheaper than the affine-invariant Riemannian metric (AIRM) since it avoids eigendecomposition. The complexity is \(O(n^3/3)\) for the Cholesky decomposition.

The Log-Cholesky distance is not affine-invariant, but it is invariant under lower triangular transformations with positive diagonal [Lin, 2019].

See also

log_cholesky_mean(): Fréchet mean under Log-Cholesky metric.
log_cholesky_geodesic(): Geodesic interpolation under Log-Cholesky metric.
airm_distance(): Distance under AIRM.
log_euclidean_distance(): Distance under Log-Euclidean metric.
bures_wasserstein_distance(): Distance under Bures-Wasserstein metric.

Examples

>>> import torch
>>> A = torch.eye(3)
>>> B = 2 * torch.eye(3)
>>> dist = log_cholesky_distance(A, B)
>>> print(f"Distance: {dist.item():.4f}")
Distance: 1.2012