Low-Rank Matrix Approximation with Weights or Missing Data Is NP-Hard

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 prove that computing an optimal weighted low-rank approximation is NP-hard, already when a rank-one approximation is sought. In fact, we show that it is hard to compute approximate solutions to the WLRA problem with some prescribed accuracy. Our proofs are based on reductions from the maximum-edge biclique problem, and apply to strictly positive weights as well as to binary weights (the latter corresponding to low-rank matrix approximation with missing data).