Low-Rank Matrix Approximation in the Infinity Norm

The low-rank matrix approximation problem with respect to the entry-wise ℓ∞-norm is the following: given a matrix M and a factorization rank r, find a matrix X whose rank is at most r and that minimizes maxi,j |Mij - Xij|. In this paper, we prove that the decision variant of this problem for r=1 is NP-complete using a reduction from the problem `not all equal 3SAT'. We also analyze several cases when the problem can be solved in polynomial time, and propose a simple practical heuristic algorithm which we apply on the problem of the recovery of a quantized low-rank matrix.