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Many real-world data sets can be presented in the form of a matrix whoseentries 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 thementioned interaction is determined by unobservable latent variables describingeach entity. Our objective is to estimate the conditional expectation of thedata matrix given the unobservable variables. This is presented as a problem ofestimation of a bivariate function referred to as graphon. We study the casesof piecewise constant and H\"older-continuous graphons. We establish finitesample risk bounds for the least squares estimator and the exponentiallyweighted aggregate. These bounds highlight the dependence of the estimationerror 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 tocompute an approximation of the least-squares estimator. Finally, we presentnumerical experiments in order to illustrate the empirical performance of thegraphon estimator on synthetic data sets.

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