Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Algorithms for Boolean Matrix Factorization using Integer Programming and Heuristics

Christos Kolomvakis, Thomas Bobille, Arnaud Vandaele, Nicolas Gillis

TL;DR

The paper tackles Boolean matrix factorization (BMF) with and without missing data, introducing IP-based alternating optimization (AO) to solve BoolLS subproblems and strategies to combine multiple BMF solutions. To address scalability, it adds scalable greedy heuristics (Greedy-BoolLS, Greedy-Comb, Greedy-TreeBMF) and a fast custom bitmatrix data structure for Boolean operations. It then demonstrates through extensive experiments on small, medium, and large datasets—including topic modeling and facial imaging—that the IP-based methods often outperform the state of the art, while the greedy approaches offer competitive accuracy with markedly better scalability. A key contribution is a practical, high-performance BMF toolkit that blends exact optimization and scalable heuristics, suitable for real-world tasks with missing data and diverse applications.

Abstract

Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. Unlike binary matrix factorization based on standard arithmetic, BMF employs the Boolean OR and AND operations for the matrix product, which improves interpretability and reduces the approximation error. It is also used in role mining and computer vision. In this paper, we first propose algorithms for BMF that perform alternating optimization (AO) of the factor matrices, where each subproblem is solved via integer programming (IP). We then design different approaches to further enhance AO-based algorithms by selecting an optimal subset of rank-one factors from multiple runs. To address the scalability limits of IP-based methods, we introduce new greedy and local-search heuristics. We also construct a new C++ data structure for Boolean vectors and matrices that is significantly faster than existing ones and is of independent interest, allowing our heuristics to scale to large datasets. We illustrate the performance of all our proposed methods and compare them with the state of the art on various real datasets, both with and without missing data, including applications in topic modeling and imaging.

Algorithms for Boolean Matrix Factorization using Integer Programming and Heuristics

TL;DR

The paper tackles Boolean matrix factorization (BMF) with and without missing data, introducing IP-based alternating optimization (AO) to solve BoolLS subproblems and strategies to combine multiple BMF solutions. To address scalability, it adds scalable greedy heuristics (Greedy-BoolLS, Greedy-Comb, Greedy-TreeBMF) and a fast custom bitmatrix data structure for Boolean operations. It then demonstrates through extensive experiments on small, medium, and large datasets—including topic modeling and facial imaging—that the IP-based methods often outperform the state of the art, while the greedy approaches offer competitive accuracy with markedly better scalability. A key contribution is a practical, high-performance BMF toolkit that blends exact optimization and scalable heuristics, suitable for real-world tasks with missing data and diverse applications.

Abstract

Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. Unlike binary matrix factorization based on standard arithmetic, BMF employs the Boolean OR and AND operations for the matrix product, which improves interpretability and reduces the approximation error. It is also used in role mining and computer vision. In this paper, we first propose algorithms for BMF that perform alternating optimization (AO) of the factor matrices, where each subproblem is solved via integer programming (IP). We then design different approaches to further enhance AO-based algorithms by selecting an optimal subset of rank-one factors from multiple runs. To address the scalability limits of IP-based methods, we introduce new greedy and local-search heuristics. We also construct a new C++ data structure for Boolean vectors and matrices that is significantly faster than existing ones and is of independent interest, allowing our heuristics to scale to large datasets. We illustrate the performance of all our proposed methods and compare them with the state of the art on various real datasets, both with and without missing data, including applications in topic modeling and imaging.

Paper Structure

This paper contains 32 sections, 13 equations, 3 figures, 12 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Boolean matrix factorization (BMF)
  3. An illustrative example
  4. Previous BMF algorithms
  5. Continuous algorithms
  6. Combinatorial algorithms
  7. Miscellaneous works
  8. IP-based algorithms for BMF
  9. IP formulation for BMF subproblems
  10. AO for BMF
  11. Initialization of AO
  12. Combining multiple BMF solutions
  13. MS-Comb-AO
  14. Tree BMF
  15. Greedy and heuristic algorithms for BMF
  16. ...and 17 more sections

Figures (3)

  • Figure 1: Venn diagram showing the overlapping communities that animals are assigned to.
  • Figure 2: Venn diagram showing the overlapping communities that animal characteristics are assigned to.
  • Figure 3: Facial features extracted by AO-BMF (left) and Greedy-TreeBMF (right) on the binarized CBCL dataset.

Theorems & Definitions (2)

  • Definition 1: Boolean matrix product
  • Definition 2: BMF