An Information Theory of Persistent Homology: Entropy, the Data Processing Inequality, and Rate–Distortion Bounds for Topological Features

Background: Topological Data Analysis (TDA) captures multi-scale geometric features of data as persistence diagrams, yet no principled information-theoretic framework quantifies how much information those features carry, how efficiently they compress, or when they are informationally irreducible. Methods: We construct a measure-theoretic probability space over persistence diagram space using a Poisson-process reference measure, and define topological entropy (H-T), topological mutual information (I-T), and a topological rate–distortion function as the core objects of a new theory. Results: Four theorems with full proofs establish finite stability, axiomatic uniqueness, a Topological Data Processing Inequality, and a Rate–Distortion Theorem with explicit Poisson-model closed-form formula. A Renyi generalization of topological entropy is also established. Computational and practical implementation aspects—including finite-sample estimation, multi-parameter extension, and algorithmic realization—are addressed inline throughout the paper. Conclusions: This framework provides a rigorous measure-theoretic information-theoretic foundation for persistent homology, demonstrated on simulated brain connectivity and point cloud data, with applications to threshold selection, genomic classification bounds, and compressed sensing.