Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Correcting matrix products over the ring of integers

Yu-Lun Wu, Hung-Lung Wang

TL;DR

The paper addresses correcting a matrix product over the ring of integers when $C$ differs from $AB$ in at most $k$ entries. It introduces a deterministic, certificate-based approach using a $m=\\sqrt{k}$ Vandermonde-inspired certificate $V$ and row/column indicators to detect and locate erroneous entries, grounded in combinatorial group testing. The resulting algorithm runs in $O(\\sqrt{k}n^2+k^2n)$ time with arithmetic bounded by $O(\\alpha^2 n^3)$, improving upon prior results for integer matrices and avoiding fast matrix multiplication. Overall, the method provides a practical, purely combinatorial alternative to randomized verification and extends Freivalds-type ideas to exact correction for integer matrices.

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 $α$ be the largest absolute value of an entry in $A$, $B$, and $C$. The integers involved in the computation are of $O(n^3α^2)$.

Correcting matrix products over the ring of integers

TL;DR

The paper addresses correcting a matrix product over the ring of integers when differs from in at most entries. It introduces a deterministic, certificate-based approach using a Vandermonde-inspired certificate and row/column indicators to detect and locate erroneous entries, grounded in combinatorial group testing. The resulting algorithm runs in time with arithmetic bounded by , improving upon prior results for integer matrices and avoiding fast matrix multiplication. Overall, the method provides a practical, purely combinatorial alternative to randomized verification and extends Freivalds-type ideas to exact correction for integer matrices.

Abstract

Let , , and be three matrices. We investigate the problem of verifying whether over the ring of integers and finding the correct product . Given that is different from by at most entries, we propose an algorithm that uses operations. Let be the largest absolute value of an entry in , , and . The integers involved in the computation are of .
Paper Structure (6 sections, 6 theorems, 9 equations, 1 figure, 2 algorithms)

This paper contains 6 sections, 6 theorems, 9 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Correcting the product
  4. The certificate
  5. The algorithm
  6. Concluding remarks

Key Result

Lemma 1

Let $p$ be a prime, and let $x_1, x_2,\dots, x_n$ be $n$ distinct integers less than $p$. For the $n\times n$ matrix $X$, with $0\leq X_{ij}<p$ and $X_{ij}\equiv x_i^{j-1} \pmod{p}$, and a column vector $\vectorbold{y}\in\mathbb{R}^n$,

Figures (1)

  • Figure 1: An illustration for the first half of Phase 2 (lines 8 to 15) of Algorithm \ref{['alg:main']}, assuming $\sqrt{k}=4$. We use the matrix $C-AB$ to illustrate the process of correction. First, an element $i$ of $S_0\cap S$ is chosen (line 9), say $i=1$. Then, row $1$ is recomputed to identify the nonzero columns, and in this example only column $4$ is identified (line 10). All nonzero entries in column $4$ are corrected, and $S_0\cap S$ is updated accordingly (lines 11 to 14). Notice that $S=\{2,3,4,5,6\}$ at the moment. The for loop stops after the second round. The remaining nonzero entries will be corrected later in lines 16 to 23.

Theorems & Definitions (13)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Remark 1
  • Remark 2
  • Lemma 4
  • proof
  • Lemma 5: Correctness of Algorithm \ref{['alg:main']}
  • ...and 3 more