# Gaussian Processes Over Graphs

## TL;DR

This paper introduces Gaussian processes tailored for signals on graphs, leveraging graph Laplacian regularization to improve prediction accuracy, especially with limited or noisy data, by enforcing specific spectral properties.

## Contribution

It develops a novel Gaussian process framework over graphs that incorporates spectral regularization, reducing predictive variance compared to traditional GPs.

## Key findings

- GPG has strictly smaller predictive variance than standard GP.
- GPG outperforms GP with small training datasets.
- GPG demonstrates robustness under noisy training conditions.

## Abstract

We propose Gaussian processes for signals over graphs (GPG) using the apriori knowledge that the target vectors lie over a graph. We incorporate this information using a graph- Laplacian based regularization which enforces the target vectors to have a specific profile in terms of graph Fourier transform coeffcients, for example lowpass or bandpass graph signals. We discuss how the regularization affects the mean and the variance in the prediction output. In particular, we prove that the predictive variance of the GPG is strictly smaller than the conventional Gaussian process (GP) for any non-trivial graph. We validate our concepts by application to various real-world graph signals. Our experiments show that the performance of the GPG is superior to GP for small training data sizes and under noisy training.