Graphon Estimation in bipartite graphs with observable edge labels and unobservable node labels

TL;DR

This paper addresses the challenge of estimating the underlying interaction patterns in bipartite graphs with unobservable node attributes by developing graphon estimation methods, providing theoretical risk bounds and practical algorithms.

Contribution

It introduces finite sample risk bounds for graphon estimators in bipartite graphs and adapts Lloyd's algorithm for practical computation.

Findings

Risk bounds depend on data size, interaction intensity, and noise level.

Proposed algorithm effectively approximates the intractable least squares estimator.

Numerical experiments demonstrate the estimator's empirical performance.

Abstract

Many real-world data sets can be presented in the form of a matrix whose entries correspond to the interaction between two entities of different natures (number of times a web user visits a web page, a student's grade in a subject, a patient's rating of a doctor, etc.). We assume in this paper that the mentioned interaction is determined by unobservable latent variables describing each entity. Our objective is to estimate the conditional expectation of the data matrix given the unobservable variables. This is presented as a problem of estimation of a bivariate function referred to as graphon. We study the cases of piecewise constant and H\"older-continuous graphons. We establish finite sample risk bounds for the least squares estimator and the exponentially weighted aggregate. These bounds highlight the dependence of the estimation error on the size of the data set, the maximum intensity of the interactions, and the level of noise. As the analyzed least-squares estimator is intractable, we propose an adaptation of Lloyd's alternating minimization algorithm to compute an approximation of the least-squares estimator. Finally, we present numerical experiments in order to illustrate the empirical performance of the graphon estimator on synthetic data sets.