Near-Optimal Column-Based Matrix Reconstruction

TL;DR

This paper introduces asymptotically optimal algorithms for low-rank matrix reconstruction using columns, leveraging fast approximate SVD-like decompositions and deterministic row selection methods.

Contribution

It presents novel algorithms that achieve near-optimal reconstruction in spectral and Frobenius norms with new deterministic row selection techniques.

Findings

Algorithms achieve asymptotic optimality in spectral and Frobenius norms.

Utilizes fast approximate SVD-like decompositions for efficient reconstruction.

Introduces deterministic algorithms based on sparse representation theorems.

Abstract

We consider low-rank reconstruction of a matrix using its columns and we present asymptotically optimal algorithms for both spectral norm and Frobenius norm reconstruction. The main tools we introduce to obtain our r esults are: (i) the use of fast approximate SVD-like decompositions for column reconstruction, and (ii) two deter ministic algorithms for selecting rows from matrices with orthonormal columns, building upon the sparse represen tation theorem for decompositions of the identity that appeared in \cite{BSS09}.