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Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations

Authors Chen Dan, Kristoffer Arnsfelt Hansen , He Jiang , Liwei Wang, Yuchen Zhou

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Author Details

Chen Dan
  • Carnegie Mellon University, Pittsburgh, Pennsylvania, United States
Kristoffer Arnsfelt Hansen
  • Department of Computer Science, Aarhus University, Aarhus, Denmark
He Jiang
  • University of Southern California, Los Angeles, California, United States
Liwei Wang
  • 1. Key Laboratory of Machine Perception, MOE, School of EECS, Peking University
  • , 2. Center for Data Science, Peking University, Beijing Institute of Big Data Research
  • , Beijing, China
Yuchen Zhou
  • Department of Statistics, University of Wisconsin-Madison, Madison, Wisconsin, United States

Funding

  • Wang, Liwei: Partially supported by National Basic Research Program of China (973 Program) (grant no. 2015CB352502).

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Chen Dan, Kristoffer Arnsfelt Hansen, He Jiang, Liwei Wang, and Yuchen Zhou. Low Rank Approximation of Binary Matrices: Column Subset Selection and Generalizations. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.MFCS.2018.41

Abstract

Low rank approximation of matrices is an important tool in machine learning. Given a data matrix, low rank approximation helps to find factors, patterns, and provides concise representations for the data. Research on low rank approximation usually focuses on real matrices. However, in many applications data are binary (categorical) rather than continuous. This leads to the problem of low rank approximation of binary matrices. Here we are given a d x n binary matrix A and a small integer k < d. The goal is to find two binary matrices U and V of sizes d x k and k x n respectively, so that the Frobenius norm of A - U V is minimized. There are two models of this problem, depending on the definition of the dot product of binary vectors: The GF(2) model and the Boolean semiring model. Unlike low rank approximation of a real matrix which can be efficiently solved by Singular Value Decomposition, we show that approximation of a binary matrix is NP-hard, even for k=1.
In this paper, our main concern is the problem of Column Subset Selection (CSS), in which the low rank matrix U must be formed by k columns of the data matrix, and we are interested in the approximation ratio achievable by CSS for binary matrices. For the GF(2) model, we show that CSS has approximation ratio bounded by k/2+1+k/(2(2^k-1)) and this is asymptotically tight. For the Boolean model, it turns out that CSS is no longer sufficient to obtain a bound. We then develop a Generalized CSS (GCSS) procedure in which the columns of U are generated from Boolean formulas operating bitwise on selected columns of the data matrix. We show that the approximation ratio achieved by GCSS is bounded by 2^(k-1)+1, and argue that an exponential dependency on k is seems inherent.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Unsupervised learning and clustering
  • Computing methodologies → Factorization methods
Keywords
  • Approximation Algorithms
  • Low Rank Approximation
  • Binary Matrices

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