spd_learn.functional.innerproduct_logchol#

spd_learn.functional.innerproduct_logchol(X, Y, Cref)[source]#

Log-Cholesky inner product.

Log-Cholesky inner product \(\mathbf{g}\) between symmetric matrices in tangent space \(\mathbf{X}\) and \(\mathbf{Y}\) at \(\mathbf{C}_\text{ref}\) is given in [1].

Parameters:

  • X (ndarray, shape (..., n, n)) – First symmetric matrices in tangent space at Cref.

  • Y (ndarray, shape (..., n, n) | None) – Second symmetric matrices in tangent space at Cref. If None, Y is set to X, giving the squared norm of X.

  • Cref (ndarray, shape (n, n)) – Reference SPD matrix.

Returns:

G – Log-Cholesky inner product between X and Y.

Return type:

float or ndarray, shape (…,)

Notes

Added in version 0.12.

See also

innerproduct

References

[1]

Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition Z. Lin. SIAM J Matrix Anal Appl, 2019, 40(4), pp. 1353-1370.