Optimal Column-Based Low-Rank Matrix Reconstruction

TL;DR

This paper establishes an optimal trade-off between the number of columns selected and the approximation quality for low-rank matrix reconstruction, providing both theoretical bounds and efficient algorithms.

Contribution

It proves an optimal approximation ratio for column-based low-rank reconstruction and introduces deterministic and randomized algorithms to find such columns efficiently.

Findings

Achieves a $ ext{sqrt}((r+1)/(r-k+1))$ approximation ratio.

Proves the trade-off is optimal up to lower order terms.

Provides algorithms with $O(r n m^{ ext{w}} ext{log} m)$ and $O(r n m^2)$ complexity.

Abstract

We prove that for any real-valued matrix $X \in \R^{m \times n}$, and positive integers $r \ge k$, there is a subset of $r$ columns of $X$ such that projecting $X$ onto their span gives a $\sqrt{\frac{r+1}{r-k+1}}$-approximation to best rank-$k$ approximation of $X$ in Frobenius norm. We show that the trade-off we achieve between the number of columns and the approximation ratio is optimal up to lower order terms. Furthermore, there is a deterministic algorithm to find such a subset of columns that runs in $O(r n m^{\omega} \log m)$ arithmetic operations where $\omega$ is the exponent of matrix multiplication. We also give a faster randomized algorithm that runs in $O(r n m^2)$ arithmetic operations.