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Graph sparsification aims to reduce the number of edges of a graph whilemaintaining its structural properties. In this paper, we propose the firstgeneral and effective information-theoretic formulation of graphsparsification, by taking inspiration from the Principle of RelevantInformation (PRI). To this end, we extend the PRI from a standard scalar randomvariable setting to structured data (i.e., graphs). Our Graph-PRI objective isachieved by operating on the graph Laplacian, made possible by expressing thegraph Laplacian of a subgraph in terms of a sparse edge selection vector$\mathbf{w}$. We provide both theoretical and empirical justifications on thevalidity of our Graph-PRI approach. We also analyze its analytical solutions ina few special cases. We finally present three representative real-worldapplications, namely graph sparsification, graph regularized multi-tasklearning, and medical imaging-derived brain network classification, todemonstrate the effectiveness, the versatility and the enhancedinterpretability of our approach over prevalent sparsification techniques. Codeof Graph-PRI is available at https://github.com/SJYuCNEL/PRI-Graphs

Principle of Relevant Information for Graph Sparsification