The Structural Complexity of Matrix-Vector Multiplication

TL;DR

This paper investigates the complexity of matrix-vector multiplication for structured matrices, showing that for matrices with low VC-dimension, the problem can be solved more efficiently, explaining practical speedups over worst-case bounds.

Contribution

The authors provide the first non-trivial upper bounds for matrix-vector multiplication on structured matrices, extending results to non-Boolean cases and applying to graph algorithms.

Findings

Efficient algorithms for Boolean matrices with low VC-dimension

Extension to non-Boolean matrices using Pollard pseudodimension

Subquadratic bounds for graph problems under structured data

Abstract

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.