Similarity-Sensitive Entropy under Representation Change and Inference
Provides foundational theory for quantifying information loss under representation changes, relevant to machine learning and information theory researchers.
The paper develops a measure-theoretic framework for similarity-sensitive entropy, proving data-processing inequalities and defining conditional entropy, while providing a counterexample to a concavity conjecture and identifying a class where concavity holds.
Similarity-sensitive entropy measures the uncertainty of a probability law relative to a similarity kernel that encodes the distinguishability between states. We develop a measure-theoretic treatment covering both finite similarity matrices and general probability spaces, and study how the law and similarity kernel transform under measurable maps, Markov kernels (channels), and conditioning operations. This yields deterministic and channel data-processing inequalities, so a reduction in entropy quantifies how much distinguishability is lost under representation change. We also define a conditional similarity sensitive entropy theory, give a counterexample to a recent conjecture on concavity, and identify a useful one-dimensional Laplace pullback class where concavity holds.