Correcting matrix products over the ring of integers

TL;DR

This paper presents an algorithm for verifying and correcting matrix products over integers when the product differs from a given matrix in at most k entries, with complexity depending on k and matrix size.

Contribution

It introduces a new algorithm that efficiently verifies and corrects matrix products over integers with bounded errors, improving over naive methods.

Findings

Algorithm uses O(√k n^2 + k^2 n) operations

Handles matrices with entries of size O(n^3 α^2)

Effective for matrices with small error k

Abstract

Let $A$, $B$, and $C$ be three $n\times n$ matrices. We investigate the problem of verifying whether $AB=C$ over the ring of integers and finding the correct product $AB$. Given that $C$ is different from $AB$ by at most $k$ entries, we propose an algorithm that uses $O(\sqrt{k}n^2+k^2n)$ operations. Let $\alpha$ be the largest absolute value of an entry in $A$, $B$, and $C$. The integers involved in the computation are of $O(n^3\alpha^2)$.