A Probabilistic Interpretation of Sampling Theory of Graph Signals
This provides a theoretical foundation for graph signal processing, potentially improving methods in semi-supervised classification, but it is incremental as it builds on existing sampling theory.
The paper tackles the problem of interpreting graph signal sampling theory probabilistically by linking it to Gaussian random field inference, showing that optimal sampling sets minimize predictive covariance in MAP estimation.
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.