On the Complexity of Robust PCA and $\ell_1$-norm Low-Rank Matrix Approximation

TL;DR

This paper proves that the problem of low-rank matrix approximation using the component-wise $ ext{L}_1$-norm, related to robust PCA, is NP-hard even in the simplest case, establishing its computational difficulty.

Contribution

The paper provides the first formal proof that $ ext{L}_1$-LRA is NP-hard, using a reduction from MAX CUT, and connects it to several other NP-hard problems.

Findings

$ ext{L}_1$-LRA is NP-hard even for rank-one matrices.

Established connections between $ ext{L}_1$-LRA and other NP-hard problems.

Provides a formal proof filling a gap in the theoretical understanding of robust PCA.

Abstract

The low-rank matrix approximation problem with respect to the component-wise $\ell_1$-norm ($\ell_1$-LRA), which is closely related to robust principal component analysis (PCA), has become a very popular tool in data mining and machine learning. Robust PCA aims at recovering a low-rank matrix that was perturbed with sparse noise, with applications for example in foreground-background video separation. Although $\ell_1$-LRA is strongly believed to be NP-hard, there is, to the best of our knowledge, no formal proof of this fact. In this paper, we prove that $\ell_1$-LRA is NP-hard, already in the rank-one case, using a reduction from MAX CUT. Our derivations draw interesting connections between $\ell_1$-LRA and several other well-known problems, namely, robust PCA, $\ell_0$-LRA, binary matrix factorization, a particular densest bipartite subgraph problem, the computation of the cut norm of $\{-1,+1\}$ matrices, and the discrete basis problem, which we all prove to be NP-hard.