pyriemann.geometry.tangentspace.transport_wasserstein¶

pyriemann.geometry.tangentspace.transport_wasserstein(X, A, B, n_steps=50)[source]¶

Parallel transport for Wasserstein metric.

The parallel transport of matrices \(\mathbf{X}\) in tangent space from an initial SPD/HPD matrix \(\mathbf{A}\) to a final SPD/HPD matrix \(\mathbf{B}\) according to the Levi-Civita connection of the Bures-Wasserstein metric, described in Section 7.5 of [1].

Bures-Wasserstein parallel transport is defined by a linear ordinary differential equation along the Wasserstein geodesic [1], integrated here with a fixed-step Runge-Kutta scheme of order 4. When \(\mathbf{A}\) and \(\mathbf{B}\) commute, a closed form is available [2].

Warning: this function must be applied to matrices \(\mathbf{X}\) already projected in tangent space with a logarithmic map at \(\mathbf{A}\), not to SPD/HPD matrices in manifold.

Parameters:

Xndarray, shape (…, n, n): Symmetric/Hermitian matrices in tangent space at A.
Andarray, shape (n, n): Initial SPD/HPD matrix.
Bndarray, shape (n, n): Final SPD/HPD matrix.
n_stepsint, default=50: Number of Runge-Kutta steps used to integrate the transport equation. Must be a positive integer. More steps are needed the further the transport map between A and B is from identity, at a cost linear in n_steps.

Returns:

X_newndarray, shape (…, n, n): Matrices in tangent space transported from A to B.