A Poincaré Inequality and Consistency Results for Signal Sampling on Large Graphs

TL;DR

A signal sampling theory for a type of graph limit -- the graphon is introduced and it is proved that complements of node subsets satisfying this inequality are unique sampling sets for Paley-Wiener spaces of graphon signals.

Abstract

Large-scale graph machine learning is challenging as the complexity of learning models scales with the graph size. Subsampling the graph is a viable alternative, but sampling on graphs is nontrivial as graphs are non-Euclidean. Existing graph sampling techniques require not only computing the spectra of large matrices but also repeating these computations when the graph changes, e.g., grows. In this paper, we introduce a signal sampling theory for a type of graph limit -- the graphon. We prove a Poincaré inequality for graphon signals and show that complements of node subsets satisfying this inequality are unique sampling sets for Paley-Wiener spaces of graphon signals. Exploiting connections with spectral clustering and Gaussian elimination, we prove that such sampling sets are consistent in the sense that unique sampling sets on a convergent graph sequence converge to unique sampling sets on the graphon. We then propose a related graphon signal sampling algorithm for large graphs, and demonstrate its good empirical performance on graph machine learning tasks.

Theorems & Definitions (7)

• proof : Proof of \ref{['thm:poincare']}
• proof : Proof of \ref{['thm:lam_uniqueness']}
• proof : Proof of \ref{['prop:uniform_map']}
• proof : Proof of \ref{['thm:component_to_uniqueness']}
• proof : Proof of \ref{['thm:consistency_general']}
• proof : Proof of \ref{['prop:sampled_points_are_uniqueness_sets']}
• proof : Proof of \ref{['thm:consistency_small']}