# Low-Rank Matrix Approximation in the Infinity Norm

## TL;DR

This paper investigates the computational complexity of low-rank matrix approximation under the entry-wise infinity norm, proving NP-completeness for rank-1 cases, analyzing solvable cases, and proposing a heuristic for practical applications.

## Contribution

It establishes NP-completeness for rank-1 approximation, analyzes polynomial-time solvable cases, and introduces a heuristic algorithm for quantized matrix recovery.

## Key findings

- Rank-1 approximation decision problem is NP-complete.
- Certain cases of the problem are solvable in polynomial time.
- A heuristic algorithm effectively recovers quantized low-rank matrices.

## Abstract

The low-rank matrix approximation problem with respect to the entry-wise $\ell_{\infty}$-norm is the following: given a matrix $M$ and a factorization rank $r$, find a matrix $X$ whose rank is at most $r$ and that minimizes $\max_{i,j} |M_{ij} - X_{ij}|$. In this paper, we prove that the decision variant of this problem for $r=1$ is NP-complete using a reduction from the problem `not all equal 3SAT'. We also analyze several cases when the problem can be solved in polynomial time, and propose a simple practical heuristic algorithm which we apply on the problem of the recovery of a quantized low-rank matrix.