  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.
