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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.02291 |
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| _version_ | 1866914655251726336 |
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| author | Stein, Dario Samuelson, Richard |
| author_facet | Stein, Dario Samuelson, Richard |
| contents | We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform, while generalizing formalisms like graphical linear algebra, convex relations and convex programming. We connect our framework to probability theory by interpreting the Laplace approximation in its context: The exactness of this approximation on normal distributions means that logdensity is a functor from Gaussian probability (densities and integration) to concave bifunctions and maximization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_02291 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory) Stein, Dario Samuelson, Richard Category Theory Optimization and Control 18M35 (Primary) 52A41 (Secondary) We introduce a compositional framework for convex analysis based on the notion of convex bifunction of Rockafellar. This framework is well-suited to graphical reasoning, and exhibits rich dualities such as the Legendre-Fenchel transform, while generalizing formalisms like graphical linear algebra, convex relations and convex programming. We connect our framework to probability theory by interpreting the Laplace approximation in its context: The exactness of this approximation on normal distributions means that logdensity is a functor from Gaussian probability (densities and integration) to concave bifunctions and maximization. |
| title | Towards a Compositional Framework for Convex Analysis (with Applications to Probability Theory) |
| topic | Category Theory Optimization and Control 18M35 (Primary) 52A41 (Secondary) |
| url | https://arxiv.org/abs/2312.02291 |