← 研踪 PaperTrailarXiv:2603.04530 · v3
Complete Diagrammatic Axiomatisations of Relative Entropy
首次提交 2026-03-05 03:15
版本更新 2026-11-03 03:45
雷达观测 2026-11-04 23:40
摘要
Relative entropy is a fundamental class of distances between probability distributions, with widespread applications in probability theory, statistics, and machine learning. In this work, we study relative entropy from a categorical perspective, viewing it as a quantitative enrichment of categories of stochastic matrices. We consider two natural monoidal structures on stochastic matrices, given by the Kronecker product and the direct sum. Our main results are complete axiomatisations of Kullback-Leibler divergence and, more generally, of Rényi divergences of arbitrary order, for each such structure. Our axiomatic theories are formulated within the framework of quantitative monoidal algebra, using a graphical language of string diagrams enriched with quantitative equations.
作者备注：Final proceedings version for MFPS XLII
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v3 · 2026-11-03 03:45 · 最新观测版本
Complete Diagrammatic Axiomatisations of Relative Entropy
Relative entropy is a fundamental class of distances between probability distributions, with widespread applications in probability theory, statistics, and machine learning. In this work, we study relative entropy from a categorical perspective, viewing it as a quantitative enrichment of categories of stochastic matrices. We consider two natural monoidal structures on stochastic matrices, given by the Kronecker product and the direct sum. Our main results are complete axiomatisations of Kullback-Leibler divergence and, more generally, of Rényi divergences of arbitrary order, for each such structure. Our axiomatic theories are formulated within the framework of quantitative monoidal algebra, using a graphical language of string diagrams enriched with quantitative equations.
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