eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-02-04
6:1
6:20
10.4230/LIPIcs.ITCS.2021.6
article
Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation
Musco, Cameron
1
Musco, Christopher
2
Woodruff, David P.
3
College of Information and Computer Sciences, University of Massachusetts Amherst, MA, USA
Department of Computer Science and Engineering, New York University Tandon School of Engineering, NY, USA
Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
In the masked low-rank approximation problem, one is given data matrix A ∈ ℝ^{n × n} and binary mask matrix W ∈ {0,1}^{n × n}. The goal is to find a rank-k matrix L for which:
cost(L) := ∑_{i=1}^n ∑_{j=1}^n W_{i,j} ⋅ (A_{i,j} - L_{i,j})² ≤ OPT + ε ‖A‖_F²,
where OPT = min_{rank-k L̂} cost(L̂) and ε is a given error parameter. Depending on the choice of W, the above problem captures factor analysis, low-rank plus diagonal decomposition, robust PCA, low-rank matrix completion, low-rank plus block matrix approximation, low-rank recovery from monotone missing data, and a number of other important problems. Many of these problems are NP-hard, and while algorithms with provable guarantees are known in some cases, they either 1) run in time n^Ω(k²/ε) or 2) make strong assumptions, for example, that A is incoherent or that the entries in W are chosen independently and uniformly at random.
In this work, we show that a common polynomial time heuristic, which simply sets A to 0 where W is 0, and then finds a standard low-rank approximation, yields bicriteria approximation guarantees for this problem. In particular, for rank k' > k depending on the public coin partition number of W, the heuristic outputs rank-k' L with cost(L) ≤ OPT + ε ‖A‖_F². This partition number is in turn bounded by the randomized communication complexity of W, when interpreted as a two-player communication matrix. For many important cases, including all those listed above, this yields bicriteria approximation guarantees with rank k' = k ⋅ poly(log n/ε).
Beyond this result, we show that different notions of communication complexity yield bicriteria algorithms for natural variants of masked low-rank approximation. For example, multi-player number-in-hand communication complexity connects to masked tensor decomposition and non-deterministic communication complexity to masked Boolean low-rank factorization.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol185-itcs2021/LIPIcs.ITCS.2021.6/LIPIcs.ITCS.2021.6.pdf
low-rank approximation
communication complexity
weighted low-rank approximation
bicriteria approximation algorithms