Approximate Multiplication of Sparse Matrices with Limited Space

TL;DR

This paper introduces a sparsity-exploiting approximate matrix multiplication method that reduces computational time while maintaining accuracy and space efficiency, suitable for large-scale applications.

Contribution

It proposes sparse co-occuring directions, an algorithm that leverages sparsity and approximate SVD to lower time complexity without sacrificing approximation quality.

Findings

Reduces time complexity to rom or large sparse matrices.

Maintains the same space complexity as previous methods.

Empirically outperforms existing algorithms in efficiency and effectiveness.

Abstract

Approximate matrix multiplication with limited space has received ever-increasing attention due to the emergence of large-scale applications. Recently, based on a popular matrix sketching algorithm -- frequent directions, previous work has introduced co-occuring directions (COD) to reduce the approximation error for this problem. Although it enjoys the space complexity of $O((m_x+m_y)\ell)$ for two input matrices $X\in\mathbb{R}^{m_x\times n}$ and $Y\in\mathbb{R}^{m_y\times n}$ where $\ell$ is the sketch size, its time complexity is $O\left(n(m_x+m_y+\ell)\ell\right)$, which is still very high for large input matrices. In this paper, we propose to reduce the time complexity by exploiting the sparsity of the input matrices. The key idea is to employ an approximate singular value decomposition (SVD) method which can utilize the sparsity, to reduce the number of QR decompositions required by COD. In this way, we develop sparse co-occuring directions, which reduces the time complexity to $\widetilde{O}\left((\nnz(X)+\nnz(Y))\ell+n\ell^2\right)$ in expectation while keeps the same space complexity as $O((m_x+m_y)\ell)$, where $\nnz(X)$ denotes the number of non-zero entries in $X$ and the $\widetilde{O}$ notation hides constant factors as well as polylogarithmic factors. Theoretical analysis reveals that the approximation error of our algorithm is almost the same as that of COD. Furthermore, we empirically verify the efficiency and effectiveness of our algorithm.