Multi-weight Matrix Completion with Arbitrary Subspace Prior Information

TL;DR

This paper introduces a multi-weight matrix completion method that leverages arbitrary subspace prior information, reducing the number of observations needed for accurate low-rank matrix recovery.

Contribution

It proposes a novel multi-weight nuclear norm minimization approach with an optimal weight selection scheme based on coherency bounds, enhancing matrix completion with prior subspace info.

Findings

Requires fewer observations than existing methods

Effectively incorporates multiple subspace priors

Improves accuracy in matrix recovery

Abstract

Matrix completion refers to completing a low-rank matrix from a few observed elements of its entries and has been known as one of the significant and widely-used problems in recent years. The required number of observations for exact completion is directly proportional to rank and the coherency parameter of the matrix. In many applications, there might exist additional information about the low-rank matrix of interest. For example, in collaborative filtering, Netflix and dynamic channel estimation in communications, extra subspace information is available. More precisely in these applications, there are prior subspaces forming multiple angles with the ground-truth subspaces. In this paper, we propose a novel strategy to incorporate this information into the completion task. To this end, we designed a multi-weight nuclear norm minimization where the weights are such chosen to penalize each angle within the matrix subspace independently. We propose a new scheme for optimally choosing the weights. Specifically, we first calculate an upper-bound expression describing the coherency of the interested matrix. Then, we obtain the optimal weights by minimizing this expression. Simulation results certify the advantages of allowing multiple weights in the completion procedure. Explicitly, they indicate that our proposed multi-weight problem needs fewer observations compared to state-of-the-art methods.