2017/10/30 by Karl Bringmann, Bringmann, Karl, Pavel Kolev +3
Engineering · Mathematics · #Data Structures and Algorithms (cs.DS) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1710.11253
openalex publication_date 2017/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the ℓ0-Low Rank Approximation Problem, where the goal is, given an m × n matrix A, to output a rank-k matrix A' for which ‖A'-A‖0 is minimized. Here, for a matrix B, ‖B‖0 denotes the number of its non-zero entries. This NP-hard variant of low rank approximation is natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions. We provide approximation algorithms which significantly improve the running time and approximation factor of previous work. For k > 1, we show how to find, in poly(mn) time for every k, a rank O(k log(n/k)) matrix A' for which ‖A'-A‖0 ≤ O(k2 log(n/k)) OPT. To the best of our knowledge, this is the first algorithm with provable guarantees for the ℓ0-Low Rank Approximation Problem for k > 1, even for bicriteria algorithms. For the well-studied case when k = 1, we give a (2+ε)-approximation in \it sublinear time, which is impossible for other variants of low rank approximation such as for the Frobenius norm. We strengthen this for the well-studied case of binary matrices to obtain a (1+O(ψ))-approximation in sublinear time, where ψ= OPT/‖ A‖0. For small ψ, our approximation factor is 1+o(1).