A Probabilistic Interpretation of Sampling Theory of Graph Signals

TL;DR

This paper provides a probabilistic perspective on graph signal sampling, linking it to Gaussian random fields and MAP inference, explaining the effectiveness of sampling strategies in semi-supervised learning.

Contribution

It introduces a probabilistic interpretation of graph sampling theory using Gaussian random fields and MAP inference, offering new insights into sampling optimality.

Findings

Sampling set with largest cut-off frequency minimizes worst-case predictive covariance.

Reconstruction via least squares is equivalent to MAP inference under certain conditions.

Probabilistic interpretation explains the success of sampling methods in semi-supervised classification.

Abstract

We give a probabilistic interpretation of sampling theory of graph signals. To do this, we first define a generative model for the data using a pairwise Gaussian random field (GRF) which depends on the graph. We show that, under certain conditions, reconstructing a graph signal from a subset of its samples by least squares is equivalent to performing MAP inference on an approximation of this GRF which has a low rank covariance matrix. We then show that a sampling set of given size with the largest associated cut-off frequency, which is optimal from a sampling theoretic point of view, minimizes the worst case predictive covariance of the MAP estimate on the GRF. This interpretation also gives an intuitive explanation for the superior performance of the sampling theoretic approach to active semi-supervised classification.