Sampling, Filtering and Sparse Approximations on Combinatorial Graphs

TL;DR

This paper develops a theoretical framework for sampling, filtering, and sparse approximation of functions on combinatorial graphs, with potential applications in image processing, data reduction, and learning.

Contribution

It introduces graph filtering using Schrödinger's operators and establishes sampling theory through Poincare and Plancherel-Polya inequalities for graphs.

Findings

Established a graph filtering method using Schrödinger operators

Proved sampling inequalities for functions on graphs

Enabled sparse approximation techniques for graph-based data

Abstract

In this paper we address sampling and approximation of functions on combinatorial graphs. We develop filtering on graphs by using Schr\"odinger's group of operators generated by combinatorial Laplace operator. Then we construct a sampling theory by proving Poincare and Plancherel-Polya-type inequalities for functions on graphs. These results lead to a theory of sparse approximations on graphs and have potential applications to filtering, denoising, data dimension reduction, image processing, image compression, computer graphics, visualization and learning theory.