spd_learn.functional.kernel_logeuclid#

spd_learn.functional.kernel_logeuclid(X, Y=None, *, Cref=None, reg=1e-10)[source]#

Log-Euclidean kernel between two sets of SPD matrices.

Log-Euclidean kernel matrix \(\mathbf{K}\) of two sets \(\mathbf{X}\) and \(\mathbf{Y}\) of SPD matrices in \(\mathbb{R}^{n \times n}\) on tangent space at \(\mathbf{C}_\text{ref}\) is calculated with pairwise inner products [1]:

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

If \(\mathbf{C}_\text{ref}\) is None [2]:

\[\mathbf{K}_{i,j} = \text{tr}(\log(\mathbf{X}_i) \log(\mathbf{Y}_j))\]

Parameters:

X (ndarray, shape (n_matrices_X, n, n)) – First set of SPD matrices.
Y (None | ndarray, shape (n_matrices_Y, n, n), default=None) – Second set of SPD matrices. If None, Y is set to X.
Cref (None | ndarray, shape (n, n), default=None) –
Reference SPD matrix. If None, Cref is defined as identity matrix.

Added in version 0.8.
reg (float, default=1e-10) – When Y is None, regularization parameter to mitigate numerical errors in kernel matrix estimation, to provide a positive-definite kernel matrix.

Returns:

K – Log-Euclidean kernel matrix between X and Y.

Return type:

ndarray, shape (n_matrices_X, n_matrices_Y)

Notes

Added in version 0.3.

Changed in version 0.8: Add parameter Cref to use a reference matrix.

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