spd_learn.functional.geodesic_chol#

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

Cholesky geodesic between SPD/HPD matrices.

The matrix at position \(\alpha\) on the Cholesky geodesic between two SPD/HPD matrices \(\mathbf{A}\) and \(\mathbf{B}\) is \(\mathbf{C} = \mathbf{L} \mathbf{L}^H\), where \(\mathbf{L}\) is computed as [1]:

\[\mathbf{L} = (1-\alpha) \text{chol}(\mathbf{A}) + \alpha \text{chol}(\mathbf{B})\]

\(\mathbf{C}\) 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 Cholesky geodesic.

Return type:

ndarray, shape (…, n, n)

Notes

Added in version 0.10.

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

See also

geodesic

References

[1]

Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging I.L. Dryden, A. Koloydenko, D. Zhou. Ann Appl Stat, 2009, 3(3), pp. 1102-1123.