spd_learn.functional.geodesic_wasserstein#

spd_learn.functional.geodesic_wasserstein(A, B, alpha=0.5)[source]#

Wasserstein geodesic between SPD/HPD matrices.

The matrix at position \(\alpha\) on the Wasserstein geodesic between two SPD/HPD matrices \(\mathbf{A}\) and \(\mathbf{B}\) is given in [1]:

\[\mathbf{C} = (1-\alpha)^2\mathbf{A} + \alpha^2\mathbf{B} + \alpha(1-\alpha)((\mathbf{AB})^{1/2} + (\mathbf{BA})^{1/2})\]

\(\mathbf{C}\) is equal to \(\mathbf{A}\) if \(\alpha\) = 0, and \(\mathbf{B}\) if \(\alpha\) = 1.

Parameters:

A (ndarray, shape (..., n, n)) – First SPD/HPD matrices.
B (ndarray, shape (..., n, n)) – Second SPD/HPD matrices.
alpha (float | ndarray, shape (...,), default=0.5) –
Position on the geodesic. If ndarray, one value per matrix pair.

Changed in version 0.12.

Returns:

C – SPD/HPD matrices on the Wasserstein geodesic.

Return type:

ndarray, shape (…, n, n)

Notes

Added in version 0.8.

Changed in version 0.12: Add support for NumPy and PyTorch. Add support for array-valued alpha.