Complete Diagrammatic Axiomatisations of Relative Entropy
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908866909831168 |
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| author | Sarkis, Ralph Zanasi, Fabio |
| author_facet | Sarkis, Ralph Zanasi, Fabio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04530 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete Diagrammatic Axiomatisations of Relative Entropy Sarkis, Ralph Zanasi, Fabio Logic in Computer Science Information Theory Category Theory 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. |
| title | Complete Diagrammatic Axiomatisations of Relative Entropy |
| topic | Logic in Computer Science Information Theory Category Theory |
| url | https://arxiv.org/abs/2603.04530 |