Fast Eigen Decomposition for Low-Rank Matrix Approximation

TL;DR

This paper introduces a fast eigen decomposition algorithm for matrices formed as weighted sums of outer products, capable of handling both positive and negative weights, improving efficiency over traditional SVD-based methods.

Contribution

The proposed algorithm efficiently computes eigen decomposition for weighted sums of outer products, including negative weights, unlike standard SVD methods.

Findings

Supports negative weights in eigen decomposition

Faster computation compared to SVD-based methods

Applicable to covariance-like matrices

Abstract

In this paper we present an efficient algorithm to compute the eigen decomposition of a matrix that is a weighted sum of the self outer products of vectors such as a covariance matrix of data. A well known algorithm to compute the eigen decomposition of such matrices is though the singular value decomposition, which is available only if all the weights are nonnegative. Our proposed algorithm accepts both positive and negative weights.