# Graph Sampling for Covariance Estimation

## TL;DR

This paper introduces methods for efficiently estimating second-order statistics of signals on graphs by subsampling vertices, enabling accurate reconstruction without spectral priors, applicable to various graph types and real-world datasets.

## Contribution

It proposes a novel subsampling approach for covariance estimation on graphs that requires fewer samples and no spectral priors, including algorithms for different models and graph types.

## Key findings

- Successful reconstruction of graph signal statistics from reduced samples.
- Development of near-optimal greedy algorithms for subsampling design.
- Validation on synthetic and real datasets demonstrating effectiveness.

## Abstract

In this paper the focus is on subsampling as well as reconstructing the second-order statistics of signals residing on nodes of arbitrary undirected graphs. Second-order stationary graph signals may be obtained by graph filtering zero-mean white noise and they admit a well-defined power spectrum whose shape is determined by the frequency response of the graph filter. Estimating the graph power spectrum forms an important component of stationary graph signal processing and related inference tasks such as Wiener prediction or inpainting on graphs. The central result of this paper is that by sampling a significantly smaller subset of vertices and using simple least squares, we can reconstruct the second-order statistics of the graph signal from the subsampled observations, and more importantly, without any spectral priors. To this end, both a nonparametric approach as well as parametric approaches including moving average and autoregressive models for the graph power spectrum are considered. The results specialize for undirected circulant graphs in that the graph nodes leading to the best compression rates are given by the so-called minimal sparse rulers. A near-optimal greedy algorithm is developed to design the subsampling scheme for the non-parametric and the moving average models, whereas a particular subsampling scheme that allows linear estimation for the autoregressive model is proposed. Numerical experiments on synthetic as well as real datasets related to climatology and processing handwritten digits are provided to demonstrate the developed theory.