We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three $n \times n$ matrices $A$, $B$, and $C$ as input, to decide whether $AB = C$. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in $\widetilde{O}(n^2)$ time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in $o(n^{\omega})$ time). To that end, we give two algorithms for MMV in the case where $AB - C$ is sparse. Specifically, when $AB - C$ has at most $O(n^{\delta})$ non-zero entries for a constant $0 \leq \delta < 2$, we give (1) a deterministic $O(n^{\omega - \varepsilon})$-time algorithm for constant $\varepsilon = \varepsilon(\delta) > 0$, and (2) a randomized $\widetilde{O}(n^2)$-time algorithm using $\delta/2 \cdot \log_2 n + O(1)$ random bits. The former algorithm is faster than the deterministic algorithm of K\"{u}nnemann (ESA, 2018) when $\delta \geq 1.056$, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses $\log_2 n + O(1)$ random bits (in turn fewer than Freivalds's algorithm). We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require $\Omega(n^{\omega})$ time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic $\widetilde{O}(n^2)$-time reductions).

翻译：我们研究矩阵乘法验证问题（MMV），其目标是在给定三个$n \times n$矩阵$A$、$B$和$C$作为输入的情况下，判定是否$AB = C$。Freivalds（MFCS, 1979）提出的一种经典随机化算法可在$\widetilde{O}(n^2)$时间内解决MMV，而一个长期挑战是（部分地）去随机化该算法，同时仍保持比矩阵乘法时间更快的运行速度（即$o(n^{\omega})$时间）。为此，针对$AB - C$为稀疏矩阵的情况，我们给出两种MMV算法。具体而言，当$AB - C$对于常数$0 \leq \delta < 2$至多有$O(n^{\delta})$个非零元素时，我们给出：(1) 一个确定性的$O(n^{\omega - \varepsilon})$时间算法，其中常数$\varepsilon = \varepsilon(\delta) > 0$；(2) 一个随机化的$\widetilde{O}(n^2)$时间算法，仅使用$\delta/2 \cdot \log_2 n + O(1)$个随机比特。当$\delta \geq 1.056$时，前一种算法比Künnemann（ESA, 2018）的确定性算法更快；后一种算法使用的随机比特少于Kimbrel和Sinha（IPL, 1993）的算法（该算法运行时间相同，但使用$\log_2 n + O(1)$个随机比特，进而少于Freivalds算法）。我们进一步研究了MMV的复杂性。首先，我们证明，对于MMV的一类自然确定性线性代数算法（包括我们的算法），所有此类算法均需$\Omega(n^{\omega})$时间。其次，我们展示了在强指数时间假设（SETH）下，为矩阵乘法（进而为MMV）证明超二次时间下界所存在的障碍。最后，我们研究了MMV的自然变体与特例（基于确定性的$\widetilde{O}(n^2)$时间归约）之间的相互关系。