# Low-rank binary matrix approximation in column-sum norm

## TL;DR

This paper introduces the first polynomial time approximation scheme for low-rank binary matrix approximation in column-sum norm over GF(2), providing near-optimal solutions efficiently.

## Contribution

It presents a randomized PTAS for $ ext{l}_1$-Rank-$r$ Approximation over GF(2), a problem previously lacking such an efficient approximation method.

## Key findings

- Achieves a $(1+ ext{epsilon})$-approximation with polynomial time complexity.
- First PTAS for $ ext{l}_1$-Rank-$r$ Approximation over GF(2).
- Provides theoretical bounds and algorithmic framework for binary matrix approximation.

## Abstract

We consider $\ell_1$-Rank-$r$ Approximation over GF(2), where for a binary $m\times n$ matrix ${\bf A}$ and a positive integer $r$, one seeks a binary matrix ${\bf B}$ of rank at most $r$, minimizing the column-sum norm $||{\bf A} -{\bf B}||_1$. We show that for every $\varepsilon\in (0, 1)$, there is a randomized $(1+\varepsilon)$-approximation algorithm for $\ell_1$-Rank-$r$ Approximation over GF(2) of running time $m^{O(1)}n^{O(2^{4r}\cdot \varepsilon^{-4})}$. This is the first polynomial time approximation scheme (PTAS) for this problem.