- Publication venue
- LIPIcs - Leibniz International Proceedings in Informatics. 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
- Publication date
- 30/11/2020
- Field of study
In maskedΒ lowβrankΒ approximation, one is given AβRnΓn and binary mask matrix Wβ{0,1}nΓn. The goal is to
find a rank-k matrix L for which: cost(L)=i=1βnβj=1βnβWi,jββ
(Ai,jββLi,jβ)2β€OPT+Ο΅β₯Aβ₯F2β,
where OPT=minrankβkΒ L^βcost(L^) and Ο΅ is a given
error parameter. Depending on the choice of W, this problem captures factor
analysis, low-rank plus diagonal decomposition, robust PCA, low-rank matrix
completion, low-rank plus block matrix approximation, and many problems. Many
of these problems are NP-hard, and while some algorithms with provable
guarantees are known, they either 1) run in time nΞ©(k2/Ο΅) or
2) make strong assumptions, e.g., that A is incoherent or that W is 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 kβ²>k depending on the $public\ coin\ partition\
numberofW,theheuristicoutputsrankβk'Lwithcost(L) \leq OPT +
\epsilon \|A\|_F^2.Thispartitionnumberisinturnboundedbytherandomized\ communication\ complexityofW,wheninterpretedasatwoβplayercommunicationmatrix.Formanyimportantexamplesofmaskedlowβrankapproximation,includingallthoselistedabove,thisresultyieldsbicriteriaapproximationguaranteeswithk' = k \cdot poly(\log n/\epsilon)$.
Further, we show that different models of communication yield 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.Comment: ITCS 202