pyriemann.utils.kernel.kernel_riemann¶

pyriemann.utils.kernel.kernel_riemann(X, Y=None, *, Cref=None, reg=1e-10)¶

Affine-invariant Riemannian kernel between two sets of SPD matrices.

Calculates the affine-invariant Riemannian kernel matrix \(\mathbf{K}\) of inner products of two sets \(\mathbf{X}\) and \(\mathbf{Y}\) of SPD matrices in \(\mathbb{R}^{n \times n}\) on tangent space at \(\mathbf{C}_\text{ref}\) by calculating pairwise products [1]:

\[\mathbf{K}_{i,j} = \text{tr}( \log( \mathbf{C}_\text{ref}^{-1/2} \mathbf{X}_i \mathbf{C}_\text{ref}^{-1/2} ) \log( \mathbf{C}_\text{ref}^{-1/2} \mathbf{Y}_j \mathbf{C}_\text{ref}^{-1/2}) )\]

Parameters:

Xndarray, shape (n_matrices_X, n, n): First set of SPD matrices.
YNone | ndarray, shape (n_matrices_Y, n, n), default=None: Second set of SPD matrices. If None, Y is set to X.
CrefNone | ndarray, shape (n, n), default=None: Reference point for the tangent space and inner product calculation. If None, Cref is calculated as the Riemannian mean of X.
regfloat, default=1e-10: Regularization parameter to mitigate numerical errors in kernel matrix estimation.

Returns:

Kndarray, shape (n_matrices_X, n_matrices_Y): The affine-invariant Riemannian kernel matrix between X and Y.