spd_learn.functional.distance_riemann#

spd_learn.functional.distance_riemann(A, B, squared=False)[source]#

Affine-invariant Riemannian distance between SPD/HPD matrices.

The affine-invariant Riemannian distance between two SPD/HPD matrices \(\mathbf{A}\) and \(\mathbf{B}\) is [1]:

\[d(\mathbf{A},\mathbf{B}) = \Vert \log(\mathbf{B}^{-1/2} \mathbf{A} \mathbf{B}^{-1/2}) \Vert_F = {\left( \sum_i \log(\lambda_i)^2 \right)}^{1/2}\]

where \(\lambda_i\) are the joint eigenvalues of \(\mathbf{A}\) and \(\mathbf{B}\).

Parameters:

A (ndarray, shape (..., n, n)) – First SPD/HPD matrices.
B (ndarray, shape (..., n, n)) – Second SPD/HPD matrices, same dimensions as A.
squared (bool, default=False) –
Return squared distance.

Added in version 0.5.

Returns:

d – Affine-invariant Riemannian distance between A and B.

Return type:

float or ndarray, shape (…,)

Notes

Changed in version 0.12: Add support for NumPy and PyTorch.