spd_learn.functional.airm_distance#

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

Compute the geodesic distance under the Affine-Invariant Riemannian Metric (AIRM).

The AIRM distance between two SPD matrices \(A\) and \(B\) is defined as the length of the geodesic connecting them:

\[d_{\text{AIRM}}(A, B) = \| \log(A^{-1/2} B A^{-1/2}) \|_F = \sqrt{\sum_{i=1}^{n} \log^2(\lambda_i)}\]

where \(\lambda_i\) are the eigenvalues of \(A^{-1/2} B A^{-1/2}\).

This metric is affine-invariant: for any invertible matrix \(W\),

\[d_{\text{AIRM}}(WAW^\top, WBW^\top) = d_{\text{AIRM}}(A, B)\]

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:

Geodesic distances with shape (…).

Return type:

torch.Tensor

Examples

>>> import torch
>>> from spd_learn.functional.metrics import airm_distance
>>> A = torch.eye(3)
>>> B = 2 * torch.eye(3)
>>> d = airm_distance(A, B)
>>> print(f"Distance: {d.item():.4f}")
Distance: 1.2012