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

Nicolas Gillis, Stephen A. Vavasis

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.

An implementation is available here: https://sites.google.com/site/nicolasgillis/code. From the page:

This Matlab code implements an exact cyclic coordinate descent method for the component-wise ℓ1-norm matrix approximation problem: Given an m-by-n matrix M and a factorization rank r, find an m-by-r matrix U and an r-by-n matrix V such that ||M-UV||1 = sumi,j |M-UV|ij is minimized. By default, it initializes the algorithm with the optimal solution of the ℓ2-norm problem using the truncated singular value decomposition provided by Matlab.

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