A Simpler Approach to Matrix Completion

TL;DR

This paper establishes improved bounds on the number of random samples needed for low-rank matrix reconstruction using nuclear norm minimization, with a simple proof leveraging quantum information theory techniques.

Contribution

It provides the best bounds to date on sampling requirements for matrix completion, simplifying the proof and connecting to quantum information theory methods.

Findings

Optimal sampling bounds for matrix completion

Reconstruction via nuclear norm minimization

Elementary proof using quantum information techniques

Abstract

This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and Oh. The reconstruction is accomplished by minimizing the nuclear norm, or sum of the singular values, of the hidden matrix subject to agreement with the provided entries. If the underlying matrix satisfies a certain incoherence condition, then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition. The proof of this assertion is short, self contained, and uses very elementary analysis. The novel techniques herein are based on recent work in quantum information theory.