Nonparametric regression on random geometric graphs sampled from submanifolds

TL;DR

This paper studies Bayesian nonparametric regression for data on unknown submanifolds, establishing minimax optimal posterior contraction rates using graph Laplacian eigenbasis under smoothness assumptions.

Contribution

It introduces a Bayesian approach leveraging random basis expansion in the graph Laplacian eigenbasis for regression on submanifolds, proving optimal convergence rates.

Findings

Posterior contraction rates are minimax optimal up to logarithmic factors.

The method applies to covariates on unknown smooth submanifolds.

The approach achieves optimal rates under Holder smoothness assumptions.

Abstract

We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.