What You Will Gain By Rounding: Theory and Algorithms for Rounding Rank

TL;DR

This paper introduces the concept of rounding rank for binary matrix factorization, exploring its theoretical properties, relationships to other areas, and evaluating algorithms for practical computation.

Contribution

It defines and studies rounding rank, connecting it to classification and dimensionality reduction, and provides an extensive experimental comparison of algorithms.

Findings

Rounding rank relates to linear classification and dimensionality reduction.

Experimental results compare algorithms for low rounding rank factorizations.

Rounding can improve binary matrix approximation with efficient algorithms.

Abstract

When factorizing binary matrices, we often have to make a choice between using expensive combinatorial methods that retain the discrete nature of the data and using continuous methods that can be more efficient but destroy the discrete structure. Alternatively, we can first compute a continuous factorization and subsequently apply a rounding procedure to obtain a discrete representation. But what will we gain by rounding? Will this yield lower reconstruction errors? Is it easy to find a low-rank matrix that rounds to a given binary matrix? Does it matter which threshold we use for rounding? Does it matter if we allow for only non-negative factorizations? In this paper, we approach these and further questions by presenting and studying the concept of rounding rank. We show that rounding rank is related to linear classification, dimensionality reduction, and nested matrices. We also report on an extensive experimental study that compares different algorithms for finding good factorizations under the rounding rank model.