spd_learn.functional.innerproduct_riemann#

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

Affine-invariant Riemannian inner product.

Affine-invariant Riemannian inner product \(\mathbf{g}\) between symmetric/Hermitian matrices in tangent space \(\mathbf{X}\) and \(\mathbf{Y}\) at \(\mathbf{C}_\text{ref}\) is [1]:

\[\mathbf{g}_{\mathbf{C}_\text{ref}}(\mathbf{X}, \mathbf{Y}) = \text{tr} \left( (\mathbf{C}_\text{ref}^{-1/2} \mathbf{X} \mathbf{C}_\text{ref}^{-1/2})^* \mathbf{C}_\text{ref}^{-1/2} \mathbf{Y} \mathbf{C}_\text{ref}^{-1/2} \right)\]

Parameters:

X (ndarray, shape (..., n, n)) – First symmetric/Hermitian matrices in tangent space at Cref.
Y (ndarray, shape (..., n, n) | None) – Second symmetric/Hermitian 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/HPD matrix.

Returns:

G – Affine-invariant Riemannian inner product between X and Y.

Return type:

float or ndarray, shape (…,)

Notes

Added in version 0.11.

Changed in version 0.12: Add support for NumPy and PyTorch.