Low-Rank Matrix Approximation with Weights or Missing Data is NP-hard

TL;DR

This paper proves that weighted low-rank matrix approximation, including cases with missing data, is NP-hard to approximate, highlighting computational challenges in data analysis applications.

Contribution

It establishes the NP-hardness of approximate weighted low-rank matrix approximation, even for rank-one cases, using reductions from the maximum-edge biclique problem.

Findings

NP-hardness of WLRA with positive weights

NP-hardness of WLRA with binary weights (missing data)

Complexity results impact data analysis and recommender systems

Abstract

Weighted low-rank approximation (WLRA), a dimensionality reduction technique for data analysis, has been successfully used in several applications, such as in collaborative filtering to design recommender systems or in computer vision to recover structure from motion. In this paper, we study the computational complexity of WLRA and prove that it is NP-hard to find an approximate solution, even when a rank-one approximation is sought. Our proofs are based on a reduction from the maximum-edge biclique problem, and apply to strictly positive weights as well as binary weights (the latter corresponding to low-rank matrix approximation with missing data).