# A divide-and-conquer algorithm for binary matrix completion

## TL;DR

This paper introduces TBMC, a divide-and-conquer algorithm for binary matrix completion that produces interpretable binary factors and outperforms existing methods on real recommender system datasets.

## Contribution

The paper presents TBMC, a novel recursive partitioning algorithm for binary matrix completion that extends PROXIMUS with missing data handling and provides a provable 2-approximation LP solution.

## Key findings

- TBMC outperforms existing methods on real datasets
- The LP approach achieves a 2-approximation for binary matrix completion
- The algorithm produces interpretable binary factors representing matrix tiles

## Abstract

We propose an algorithm for low rank matrix completion for matrices with binary entries which obtains explicit binary factors. Our algorithm, which we call TBMC (\emph{Tiling for Binary Matrix Completion}), gives interpretable output in the form of binary factors which represent a decomposition of the matrix into tiles. Our approach is inspired by a popular algorithm from the data mining community called PROXIMUS: it adopts the same recursive partitioning approach while extending to missing data. The algorithm relies upon rank-one approximations of incomplete binary matrices, and we propose a linear programming (LP) approach for solving this subproblem. We also prove a $2$-approximation result for the LP approach which holds for any level of subsampling and for any subsampling pattern. Our numerical experiments show that TBMC outperforms existing methods on recommender systems arising in the context of real datasets.