Accelerated decomposition of bistochastic kernel matrices by low rank approximation

TL;DR

This paper introduces a fast algorithm for approximating eigenvalue decompositions of bistochastic kernel matrices using low rank approximations, significantly reducing computational costs for large datasets.

Contribution

It presents a novel accelerated method employing pivoted partial Cholesky for efficient low rank approximation without full matrix formation.

Findings

Achieves linear complexity in dataset size

Maintains high accuracy in extracting spatiotemporal patterns

Outperforms subsampling and Nystroem extension methods

Abstract

We develop an accelerated algorithm for computing an approximate eigenvalue decomposition of bistochastic normalized kernel matrices. 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 bistochastic 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 apply the proposed algorithm to the kernel based extraction of spatiotemporal patterns from chaotic dynamics, demonstrating its accuracy while also comparing it with an alternative algorithm consisting of subsampling and Nystroem extension.