pyriemann.utils.geodesic.geodesic_wasserstein

pyriemann.utils.geodesic.geodesic_wasserstein(A, B, alpha=0.5)

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:

Andarray, shape (…, n, n)

First SPD/HPD matrices.

Bndarray, shape (…, n, n)

Second SPD/HPD matrices.

alphafloat, default=0.5

Position on the geodesic.

Returns:

Cndarray, shape (…, n, n)

SPD/HPD matrices on the Wasserstein geodesic.

See also

geodesic

Notes

..versionadded:: 0.8

References

[1]

Wasserstein Riemannian geometry of Gaussian densities L. Malagò, L. Montrucchio, G. Pistone. Information Geometry, 2018, 1, pp. 137–179.