# Low-rank approximation in the Frobenius norm by column and row subset selection

## TL;DR

This paper introduces a fast, robust deterministic algorithm for low-rank matrix approximation using column and row subset selection, achieving error bounds close to the optimal Frobenius norm approximation, and extends the method to tensor approximations.

## Contribution

It presents a novel, efficient deterministic algorithm for column and row subset selection with provable error bounds, improving upon previous randomized methods and extending to tensor approximations.

## Key findings

- Algorithm guarantees error close to best Frobenius norm approximation.
- Error bound within a factor of k+1 of optimal for cross approximation.
- First polynomial-time deterministic algorithm with such error bounds.

## Abstract

A CUR approximation of a matrix $A$ is a particular type of low-rank approximation $A \approx C U R$, where $C$ and $R$ consist of columns and rows of $A$, respectively. One way to obtain such an approximation is to apply column subset selection to $A$ and $A^T$. In this work, we describe a numerically robust and much faster variant of the column subset selection algorithm proposed by Deshpande and Rademacher, which guarantees an error close to the best approximation error in the Frobenius norm. For cross approximation, in which $U$ is required to be the inverse of a submatrix of $A$ described by the intersection of $C$ and $R$, we obtain a new algorithm with an error bound that stays within a factor $k + 1$ of the best rank-$k$ approximation error in the Frobenius norm. To the best of our knowledge, this is the first deterministic polynomial-time algorithm for which this factor is bounded by a polynomial in $k$. Our derivation and analysis of the algorithm is based on derandomizing a recent existence result by Zamarashkin and Osinsky. To illustrate the versatility of our new column subset selection algorithm, an extension to low multilinear rank approximations of tensors is provided as well.