Approximation Schemes for Low-rank Binary Matrix Approximation Problems

Abstract

We provide a randomized linear time approximation scheme for a generic problem about clustering of binary vectors subject to additional constrains. The new constrained clustering problem encompasses a number of problems and by solving it, we obtain the first linear timeapproximation schemes for a number of well-studied fundamental problems concerning clustering of binary vectors and low-rank approximation of binary matrices. Among the problems solvable by our approach are Low GF(2)-Rank Approximation, Low Boolean-Rank Approximation, and various versions of Binary Clustering. For example, for Low GF(2)-Rank Approximation problem, where for an m × n binary matrix A and integer r > 0, we seek for a binary matrix B of GF(2) rank at most r such that 0 norm of matrix A − B is minimum, our algorithm, for any > 0 in time f (r, ) · n · m, where f is some computable function, outputs a (1+ )-approximate solution with probability at least (1− 1 e ). Our approximation algorithms substantially improve the running times and approximation factors of previous works. We also give (deterministic) PTASes for these problems running in time n f (r) 1 2 log 1 , where f is some function depending on the problem. Our algorithm for the constrained clustering problem is based on a novel sampling lemma, which is interesting in its own. * The research leading to these results have been supported by the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL".

On the Parameterized Complexity of Clustering Incomplete Data into Subspaces of Small Rank

“…Related results are also presented in very recent works which investigated the complexity of different matrix editing and clustering problems from the parameterized and approximation perspectives (Fomin, Golovach, and Panolan 2018;Fomin et al 2019;Eiben et al 2019). All of these papers except for the last one are concerned with the complete data setting.…”

Parameterized algorithms for identifying gene co-expression modules via weighted clique decomposition

“…The non-symmetric variants of these matrix problems known as Binary Matrix Factorization, have been receiving a lot of attention recently [25,15,16,1,19,3]. Here the objective is to minimize A − BC , where A is an m × n input matrix, B is an m × k output binary matrix and C is a k×n output binary matrix.…”

Recent Developments in Boolean Matrix Factorization

“…Furthermore, Fomin et al (2019) provided an algorithm which computes a (1 + ε)approximate BMF in time 2 2 O(k) /ε 2 •lg 2 (1/ε) • mn. Note that when k and ε are constants, then the running time of this algorithm becomes O(mn), i.e.…”

“…Bhattacharya et al (2019) extended ideas of Fomin et al (2019) and Ban et al (2019) to obtain a 4-pass streaming algorithm which computes a (1 + ε)-approximate BMF. Their algorithm never stores more than 2Õ (2 k /ε 2 ) • (lg n) 2k rows of the matrix and it has running time 2Õ (2 k /ε 2 ) • (lg n) 2k • mn.…”

“…More recently, Miettinen et al (2006) introduced the problem to the data mining community, Vaidya et al (2007) to the access control community, Bělohlávek and Vychodil (2010) to formal concept analysts, and van den Broeck and Darwiche (2013) to lifted inference community, to name but a few examples. Even more recently, Ravanbakhsh et al (2016) studied the problem in the framework of machine learning, increasing the interest of that community, and Ban et al (2019); Chandran et al (2016); Fomin et al (2019) (re-)launched the interest in the problem in the theory community.…”