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

• URL: http://arxiv.org/abs/2311.10610v3
• Date: Wed, 09 Oct 2024 16:28:15 GMT
• Title: A Poincaré Inequality and Consistency Results for Signal Sampling on Large Graphs
• Authors: Thien Le, Luana Ruiz, Stefanie Jegelka,
• Abstract summary: We introduce a signal sampling theory for a type of graph limit -- the graphon. We show 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.
• Score: 34.99659089854587
• License:
• 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 Poincar\'e 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.

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