Correcting matrix products over the ring of integers

Designing efficient algorithms to multiply two matrices is an ongoing research topic, due to its wide applications. The research originates from Strassen [9] in 1969, and the known fastest algorithm was proposed by Duan, Wu, and Zhou [2] in 2023, with time complexity $O(n^{2.37188})$, where $n$ is the row- and column-dimensions of the matrices. Although the best known algorithm for matrix multiplication is still super-quadratic, verifying whether the product of two matrices is equal to a given matrix can be done more efficiently. Freivalds [3] proposed a randomized algorithm that verifies with high probability whether $AB=C$, for given matrices $A$, $B$, and $C$. In Freivalds’ algorithm, a vector $\mathbf{v}$ is chosen at random, and the result of whether $A(B\mathbf{v})=C\mathbf{v}$ is reported. Clearly, Freivalds’ algorithm has one-sided error; namely the answer may be false-positive. Nevertheless, there are only $O(n^{2})$ operations needed.

It is still unknown whether Freivalds’ algorithm can be efficiently derandomized [4]. An immediate attempt for derandomizing Freivalds’ algorithm is to choose a vector $\mathbf{x}$ of the form $(1,x,x^{2},\dots,x^{n})$. Then $AB\mathbf{x}-C\mathbf{x}$ consists of $n$ polynomials in $x$. The problem reduces to determining whether the nonzero polynomials are all zero at ${x}$. If $x$ is chosen as a non-root to all nonzero polynomials, then the modified algorithm verifies whether $AB=C$ deterministically. However, even though such an $x$ can be found, for example, by applying Cauchy’s bound, one needs to manipulate numbers which are exponentially large. This makes the algorithm impractical. Related discussion can be found in [4, 5].

Recently, Gąsieniec et al. [4] showed that, if the matrix $C$ has at most $k$ erroneous entries, then these entries can be identified and also corrected in deterministic $\tilde{O}(kn^{2})$ time, where $\tilde{O}(\cdot)$ suppresses polylogarithmic terms in $n$ and $k$. Their approach resembles methods from combinatorial group testing [1]. Based on the same concept, randomized algorithms were also developed for $k$ equal to the number of erroneous entries. When the number of erroneous entries is not known, they applied the technique of compressed matrix multiplication [8], and developed a randomized algorithm which runs in $\tilde{O}(n^{2}+kn)$ time. Note that the algorithms of Gąsieniec et al. applies for matrix multiplications over any ring. For matrix multiplications over the ring of integers, Kutzkov [7] developed a deterministic algorithm running in $O(n^{2}+k^{2}n\log^{5}n)$ time; Künnemann [6] proposed an ${O}(\sqrt{k}n^{2}\log^{2+o(1)}n+k^{2}\log^{3+o(1)}n)$-time deterministic algorithm, given that the absolute values of the input integers are upper bounded by $n^{c}$ for some constant $c$.

In this paper, we are also concerned with the problem of correcting $C$ to be the product of $A$ and $B$, where $A$, $B$, and $C$ are three $n\times n$ matrices with $C$ different from $AB$ by at most $k$ entries. The matrix multiplication is restricted to be over the ring of integers. We develop a deterministic algorithm, which runs in $O(\sqrt{k}n^{2}+k^{2}n)$ time. Our algorithm is purely combinatorial; i.e. it does not rely on fast matrix multiplications or other well-developed subroutines. The idea is inherited from Gąsieniec et al. [4], based on combinatorial group testing. This can also be viewed as an extension of Freivalds’ algorithm; a number of vectors $\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{m}$ are chosen, and the erroneous entries are identified and corrected based on the result of $(AB-C)(\mathbf{x}_{1}\mid\mathbf{x}_{2}\mid\dots\mid\mathbf{x}_{m})$.

We note here that the values manipulated by the algorithm are of $O(\alpha^{2}n^{3})$, where $\alpha$ is the largest absolute value of an entry in the input matrices. All values to manipulate are in a reasonable range, namely polynomial in both $n$ and the input values. In the remainder of this paper, we summarize necessary tools in Section 2, and then give the algorithm in Section 3.

For a matrix $A$, we use $A_{ij}$ to denote the entry at the intersection of the $i$th row and the $j$th column. We use $[n]$ to denote the set $\{x\in\mathbb{N}\colon\,1\leq x\leq n\}$. Let $x_{1},x_{2},\dots,x_{n}$ be $n$ integers. The $n\times n$ Vandermonde matrix of $x_{1},x_{2},\dots,x_{n}$ is defined to be the matrix $V$ with $V_{ij}=x_{i}^{j-1}$; i.e.

In the following, we use $n$ to denote the row- and column-dimensions of the input matrices, and we let $p$ be a prime number greater than $n$.

In this paper, we investigate the problem of correcting matrix products over the ring of integers. We propose a simple deterministic $O(\sqrt{k}n^{2}+k^{2}n)$-time algorithm, where $n$ is the row- and column-dimensions, and $k$ is an upper bound on the number of erroneous entries. When $k<n$, the proposed algorithm is more efficient than the naive $\Theta(n^{3})$-time algorithm to multiply two $n\times n$ matrices. For certain $k$, it also slightly improves the existing ones, e.g. [4, 6, 7]. In addition, the values involved during the execution are in a reasonable range, namely polynomial in $n$ and the input values, so the algorithm can be implemented with the requested time complexity on conventional computation models.

The authors thank the anonymous reviewers, whose comments help improving the presentation, and also thank Wing-Kai Hon and Meng-Tsung Tsai for the discussion regarding the computation models. This research was supported in part by the Ministry of Science and Technology of Taiwan under contract MOST grant 110-2221-E-003-003-MY3.