# Predicting Graph Signals using Kernel Regression where the Input Signal is Agnostic to a Graph

## TL;DR

This paper introduces a kernel regression approach for predicting signals on a graph from potentially unrelated input data, incorporating graph regularization and learning the graph structure when unknown, demonstrating robustness with limited and noisy data.

## Contribution

It presents a novel kernel regression framework that handles agnostic inputs and learns the underlying graph structure, extending graph signal prediction capabilities.

## Key findings

- Effective noise reduction and smoothing in predictions.
- Robust performance with limited and noisy training data.
- Good reconstruction even with highly under-determined sampling.

## Abstract

We propose a kernel regression method to predict a target signal lying over a graph when an input observation is given. The input and the output could be two different physical quantities. In particular, the input may not be a graph signal at all or it could be agnostic to an underlying graph. We use a training dataset to learn the proposed regression model by formulating it as a convex optimization problem, where we use a graph-Laplacian based regularization to enforce that the predicted target is a graph signal. Once the model is learnt, it can be directly used on a large number of test data points one-by-one independently to predict the corresponding targets. Our approach employs kernels between the various input observations, and as a result the kernels are not restricted to be functions of the graph adjacency/Laplacian matrix. We show that the proposed kernel regression exhibits a smoothing effect, while simultaneously achieving noise-reduction and graph-smoothness. We then extend our method to the case when the underlying graph may not be known apriori, by simultaneously learning an underlying graph and the regression coefficients. Using extensive experiments, we show that our method provides a good prediction performance in adverse conditions, particularly when the training data is limited in size and is noisy. In graph signal reconstruction experiments, our method is shown to provide a good performance even for a highly under-determined subsampling.