Accelerated decomposition of bistochastic kernel matrices by low rank approximation

Abstract

We develop an accelerated algorithm for the approximate eigenvalue decomposition of symmetrically normalized kernel matrices, focusing on a bistochastic normalization. Our approach constructs a low rank approximation of the original kernel matrix by the pivoted partial Cholesky algorithm, and uses it to compute an approximate decomposition of its normalization without requiring the formation of the full kernel matrix. The cost of the proposed algorithm depends linearly on the size of the employed training dataset and quadratically on the rank of the low rank approximation, offering a significant cost reduction compared to the naive approach. We derive trace norm error bounds for the approximation of two classes of normalized kernel matrices. We apply the proposed algorithm to the kernel based extraction of spatiotemporal patterns from chaotic Kuramoto-Sivashinsky dynamics.