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，判定是否满足 $AB = C$。Freivalds（MFCS, 1979）提出的经典随机算法可在 $\widetilde{O}(n^2)$ 时间内求解 MMV，而一个长期存在的挑战是在运行时间仍快于矩阵乘法时间（即 $o(n^{\omega})$ 时间）的前提下，（部分地）实现该算法的去随机化。为此，我们针对 $AB - C$ 为稀疏矩阵的情形给出了两种 MMV 算法。具体而言，当 $AB - C$ 至多有 $O(n^{\delta})$ 个非零元素（其中常数 $0 \leq \delta < 2$）时，我们给出：（1）一个确定性 $O(n^{\omega - \varepsilon})$ 时间算法，其中常数 $\varepsilon = \varepsilon(\delta) > 0$；（2）一个随机 $\widetilde{O}(n^2)$ 时间算法，其使用 $\delta/2 \cdot \log_2 n + O(1)$ 随机比特。当前者算法在 $\delta \geq 1.056$ 时，比 K\"{u}nnemann（ESA, 2018）的确定性算法更快；后者算法与 Kimbrel 和 Sinha（IPL, 1993）的算法运行时间相同，但使用的随机比特数更少（为 $\log_2 n + O(1)$ 比特，这本身已少于 Freivalds 算法）。此外，我们研究了 MMV 的复杂度。我们首先证明，针对 MMV 的一类自然确定性线性代数算法（包括我们的算法）均需要 $\Omega(n^{\omega})$ 时间。我们还展示了在强指数时间假设（SETH）下，证明矩阵乘法（进而 MMV）具有超二次运行时间下界的障碍。最后，我们研究了 MMV 的自然变体与特殊情形之间的关联（通过确定性 $\widetilde{O}(n^2)$ 时间归约）。