Adaptive estimation of nonparametric geometric graphs

Probabilités et Statistique

Lieu: 
Salle séminaire M3-324
Orateur: 
Thanh Mai Pham Ngoc
Affiliation: 
Université Paris-Sud Orsay
Dates: 
Mercredi, 19 Février, 2020 - 10:30 - 11:30
Résumé: 

This talks studies the recovery of graphons when they are convolution kernels on compact (symmetric) metric spaces. This case is of particular interest since it covers the situation where the probability of an edge depends only on some unknown nonparametric function of the distance between latent points, referred to as Nonparametric Geometric Graphs (NGG). 
In this setting, adaptive estimation of NGG is possible using a spectral procedure combined with a Goldenshluger-Lepski adaptation method. The latent spaces covered by our framework encompass (among others) compact symmetric spaces of rank one, namely real spheres and projective spaces. For these latter, explicit computations of the eigen-basis and of the model complexity can be achieved, leading to quantitative non-asymptotic results. The time complexity of our method scales cubicly in the size of the graph and exponentially in the regularity of the graphon. Hence, this paper offers an algorithmically and theoretically efficient procedure to estimate smooth NGG. As a by product, this paper shows a non-asymptotic concentration result on the spectrum of integral operators defined by symmetric kernels (not necessarily positive).