Estimating a smooth function on a large graph by Bayesian Laplacian regularisation

TL;DR

This paper develops a Bayesian Laplacian regularization method for estimating smooth functions on large graphs, providing theoretical guarantees of asymptotic optimality under certain graph and function smoothness assumptions.

Contribution

It introduces a Bayesian approach with Laplacian-based Gaussian priors for graph-based function estimation, along with theoretical analysis of its asymptotic optimality.

Findings

Achieves asymptotically optimal Bayesian regularization.

Provides theoretical results linking graph Laplacian properties to estimation accuracy.

Validates the approach under specific graph shape and smoothness conditions.

Abstract

We study a Bayesian approach to estimating a smooth function in the context of regression or classification problems on large graphs. We derive theoretical results that show how asymptotically optimal Bayesian regularization can be achieved under an asymptotic shape assumption on the underlying graph and a smoothness condition on the target function, both formulated in terms of the graph Laplacian. The priors we study are randomly scaled Gaussians with precision operators involving the Laplacian of the graph.