We consider the problem of preprocessing an $n\times n$ matrix $\mathbf{M}$, and supporting queries that, for any vector $v$, returns the matrix-vector product $\mathbf{M} v$. This problem has been extensively studied in both theory and practice: on one side, practitioners have developed algorithms that are highly efficient in practice, whereas on the other side, theoreticians have proven that the problem cannot be solved faster than naive multiplication in the worst-case. This lower bound holds even in the average-case, implying that existing average-case analyses cannot explain this gap between theory and practice. Hence, we study the problem for \emph{structured} matrices. We show that for $n\times n$ Boolean matrices of VC-dimension $d$, the matrix-vector multiplication problem can be solved with $\widetilde{O}(n^2)$ preprocessing and $\widetilde{O}(n^{2-1/d})$ query time. Given the low constant VC-dimensions observed in most real-world data, our results posit an explanation for why the problem can be solved so much faster in practice. Furthermore, we show how to extend this result to the non-Boolean setting with the Pollard pseudodimension. Our results yield the first non-trivial upper bounds for many applications. In previous works, the online matrix-vector (OMv) hypothesis (conjecturing that quadratic time is needed per query, even over the boolean semi-ring) was used to prove many conditional lower bounds, showing that it is impossible to compute and maintain high-accuracy estimates for effective resistance, Laplacian solvers, shortest paths, and triangle detection in graphs subject to node insertions and deletions in subquadratic time. Yet, via a reduction to our matrix-vector multiplication result, we show we can maintain these problems efficiently if the input is structured, providing the first subquadratic upper bounds in the high-accuracy regime.

翻译：我们研究预处理一个 $n\times n$ 矩阵 $\mathbf{M}$ 并支持查询的问题：对于任意向量 $v$，返回矩阵-向量乘积 $\mathbf{M} v$。该问题在理论和实践中均得到了广泛研究：一方面，实践者开发了在实践中非常高效的算法；另一方面，理论家已证明在最坏情况下，该问题无法比朴素乘法更快求解。这一下界甚至在平均情况下也成立，这意味着现有的平均情况分析无法解释理论与实践之间的这种差距。因此，我们针对\emph{结构化}矩阵研究该问题。我们证明，对于 VC 维为 $d$ 的 $n\times n$ 布尔矩阵，矩阵-向量乘法问题可通过 $\widetilde{O}(n^2)$ 的预处理时间和 $\widetilde{O}(n^{2-1/d})$ 的查询时间求解。鉴于在大多数现实世界数据中观察到的 VC 维常数较低，我们的结果为该问题在实践中能够更快求解提供了一种解释。此外，我们展示了如何利用 Pollard 伪维将这一结果推广到非布尔设置。我们的结果为许多应用提供了首个非平凡上界。在先前的工作中，在线矩阵-向量（OMv）假设（推测即使在布尔半环上，每次查询也需要二次时间）被用于证明许多条件性下界，表明在亚二次时间内无法计算和维护有效电阻、拉普拉斯求解器、最短路径以及图中节点插入和删除下的三角形检测的高精度估计。然而，通过归约到我们的矩阵-向量乘法结果，我们证明如果输入是结构化的，则可以高效维护这些问题，从而在高精度机制下提供了首个亚二次上界。