spd_learn.functional.geodesic_riemann#

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

Affine-invariant Riemannian geodesic between SPD/HPD matrices.

The matrix at position \(\alpha\) on the affine-invariant Riemannian geodesic between two SPD/HPD matrices \(\mathbf{A}\) and \(\mathbf{B}\) is:

\[\mathbf{C} = \mathbf{A}^{1/2} \left( \mathbf{A}^{-1/2} \mathbf{B} \mathbf{A}^{-1/2} \right)^\alpha \mathbf{A}^{1/2}\]

\(\mathbf{C}\) 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 affine-invariant Riemannian geodesic.

Return type:

ndarray, shape (…, n, n)

Notes

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