Optimal CUR Matrix Decompositions

TL;DR

This paper introduces efficient algorithms for CUR matrix decompositions that approximate a given matrix with provable guarantees, using fewer columns and rows while maintaining near-optimal rank and approximation quality.

Contribution

It presents input-sparsity-time and deterministic algorithms for CUR decomposition with optimal parameters, improving computational efficiency and approximation guarantees.

Findings

Algorithms run in input-sparsity time

Achieve near-optimal column and row subset sizes

Guarantee approximation within (1+ε) of best rank-k approximation

Abstract

The CUR decomposition of an $m \times n$ matrix $A$ finds an $m \times c$ matrix $C$ with a subset of $c < n$ columns of $A,$ together with an $r \times n$ matrix $R$ with a subset of $r < m$ rows of $A,$ as well as a $c \times r$ low-rank matrix $U$ such that the matrix $C U R$ approximates the matrix $A,$ that is, $ || A - CUR ||_F^2 \le (1+\epsilon) || A - A_k||_F^2$, where $||.||_F$ denotes the Frobenius norm and $A_k$ is the best $m \times n$ matrix of rank $k$ constructed via the SVD. We present input-sparsity-time and deterministic algorithms for constructing such a CUR decomposition where $c=O(k/\epsilon)$ and $r=O(k/\epsilon)$ and rank$(U) = k$. Up to constant factors, our algorithms are simultaneously optimal in $c, r,$ and rank$(U)$.