Sublinear randomized algorithms for skeleton decompositions

TL;DR

This paper presents a sublinear randomized algorithm for skeleton decomposition of matrices, providing error guarantees and analyzing existing algorithms, with applications in efficient matrix approximation.

Contribution

It introduces a novel sublinear randomized method for skeleton decomposition with theoretical error bounds and analyzes existing algorithms within this framework.

Findings

Algorithm runs in O(l^3) time with high probability error bounds.

Sampling approximately k log n rows and columns suffices for accurate approximation.

Regularization is essential for nonsymmetric matrices to achieve good results.

Abstract

Let $A$ be a $n$ by $n$ matrix. A skeleton decomposition is any factorization of the form $CUR$ where $C$ comprises columns of $A$, and $R$ comprises rows of $A$. In this paper, we consider uniformly sampling $\l\simeq k \log n$ rows and columns to produce a skeleton decomposition. The algorithm runs in $O(\l^3)$ time, and has the following error guarantee. Let $\norm{\cdot}$ denote the 2-norm. Suppose $A\simeq X B Y^T$ where $X,Y$ each have $k$ orthonormal columns. Assuming that $X,Y$ are incoherent, we show that with high probability, the approximation error $\norm{A-CUR}$ will scale with $(n/\l)\norm{A-X B Y^T}$ or better. A key step in this algorithm involves regularization. This step is crucial for a nonsymmetric $A$ as empirical results suggest. Finally, we use our proof framework to analyze two existing algorithms in an intuitive way.