Low-Rank Matrix Approximation in the Infinity Norm
This addresses a computational complexity problem in matrix approximation for researchers in optimization and machine learning, but it is incremental as it builds on existing low-rank approximation work.
The paper tackles the low-rank matrix approximation problem under the infinity norm, proving NP-completeness for rank 1 and identifying polynomial-time solvable cases, with a heuristic applied to quantized low-rank matrix recovery.
The low-rank matrix approximation problem with respect to the entry-wise $\ell_{\infty}$-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 $\max_{i,j} |M_{ij} - X_{ij}|$. 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.