Adaptive Estimation of Nonparametric Geometric Graphs

TL;DR

This paper introduces an adaptive spectral method for estimating nonparametric geometric graphs, covering a broad class of latent spaces, with theoretical guarantees and practical efficiency for smooth graphons.

Contribution

It develops a novel spectral estimation procedure combined with a Goldenshluger-Lepski method for adaptive recovery of NGG on compact symmetric spaces.

Findings

Provides non-asymptotic bounds for spectral concentration.

Achieves explicit eigen-basis computations for spheres and projective spaces.

Offers an algorithm with cubic time complexity in graph size.

Abstract

This article 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).