pyriemann.geometry.tangentspace.innerproduct_riemann¶

pyriemann.geometry.tangentspace.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:

Xndarray, shape (…, n, n): First symmetric/Hermitian matrices in tangent space at Cref.
Yndarray, 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.
Crefndarray, shape (n, n): Reference SPD/HPD matrix.

Returns:

Gfloat or ndarray, shape (…,): Affine-invariant Riemannian inner product between X and Y.