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Compositional statistical mechanics, entropy and variational inference
Published 24 Mar 2024 in math-ph, math.MP, and math.PR | (2403.16104v1)
Abstract: In this document, we aim to gather various results related to a compositional/categorical approach to rigorous Statistical Mechanics. Rigorous Statistical Mechanics is centered on the mathematical study of statistical systems. Central concepts in this field have a natural expression in terms of diagrams in a category that couples measurable maps and Markov kernels. We showed that statistical systems are particular representations of partially ordered sets (posets), that we call A-specifications, and expressed their phases, i.e., Gibbs measures, as invariants of these representations. It opens the way to the use of homological algebra to compute phases of statistical systems. Two central results of rigorous Statistical Mechanics are, firstly, the characterization of extreme Gibbs measures as it relates to the zero-one law for extreme Gibbs measures, and, secondly, their variational principle which states that for translation invariant Hamiltonians, Gibbs measures are the minima of the Gibbs free energy. We showed how the characterization of extreme Gibbs measures extends to A-specifications; we proposed an Entropy functional for A-specifications and gave a message-passing algorithm, that generalized the belief propagation algorithm of graphical models, to find critical points of the associated variational free energy.
- G. Sergeant-Perthuis, “Compositional statistical mechanics, entropy and variational inference,” in Twelfth Symposium on Compositional Structures, 2024.
- G. Sergeant-Perthuis, “Bayesian/graphoid intersection property for factorisation spaces,” 2021. https://arxiv.org/abs/1903.06026.
- G. Sergeant-Perthuis, “Intersection property and interaction decomposition,” Apr. 2019. https://arxiv.org/abs/1904.09017.
- G. Sergeant-Perthuis, “Interaction decomposition for presheaves,” Aug. 2020. https://arxiv.org/abs/2008.09029.
- D. Bennequin, O. Peltre, G. Sergeant-Perthuis, and J. P. Vigneaux, “Extra-fine sheaves and interaction decompositions,” Sept. 2020. https://arxiv.org/abs/2009.12646.
- G. Sergeant-Perthuis, “Interaction decomposition for Hilbert spaces,” 2021. https://arxiv.org/abs/2107.06444.
- PhD thesis, Université de Paris, 2021. Link.
- G. Sergeant-Perthuis, “A categorical approach to statistical mechanics,” in Geometric Science of Information (F. Nielsen and F. Barbaresco, eds.), (Cham), pp. 258–267, Springer Nature Switzerland, 2023.
- G. Sergeant-Perthuis, “Characterization of extreme Gibbs measures for a Categorical Approach to Statistical Mechanics,” Feb. 2024. https://hal.sorbonne-universite.fr/hal-04456412.
- G. Sergeant-Perthuis, “Regionalized optimization,” 2022. https://arxiv.org/abs/2201.11876.
- O. Peltre, “Message passing algorithms and homology,” 2020. Ph.D. thesis, Link.
- J. Yedidia, W. Freeman, and Y. Weiss, “Constructing Free-Energy Approximations and Generalized Belief Propagation Algorithms,” IEEE Transactions on Information Theory, vol. 51, pp. 2282–2312, July 2005.
- D. Ruelle, Statistical Mechanics. Imperial College Press, 1999.
- MIT Press, 2006.
- Imperial College Press, 2010.
- No. 86 in Springer series in chemical physics, Berlin ; New York: Springer, 2007. OCLC: ocm79447449.
- Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.
- USA: Oxford University Press, Inc., 2009.
- M. Gromov, “In a search for a structure, part 1: On entropy.,” 2013.
- T. Fritz, “A synthetic approach to markov kernels, conditional independence and theorems on sufficient statistics,” Advances in Mathematics, 2020.
- J. P. Vigneaux, “Information structures and their cohomology,” Theory and Applications of Categories, vol. 35, no. 38, pp. 1476–1529, 2020.
- T. Fritz and D. I. Spivak, “Internal probability valuations.” Workshop Categorical Probability and Statistics, 2020.
- J.-C. Belfiore and D. Bennequin, “Topos and stacks of deep neural networks,” ArXiv, 2021.
- T. Fritz and P. Perrone, “A probability monad as the colimit of spaces of finite samples,” Theory and Applications of Categories, 2019.
- M. Marcolli, “Gamma spaces and information,” Journal of Geometry and Physics, 2019.
- P. Baudot and D. Bennequin, “The homological nature of entropy,” Entropy, vol. 17, no. 5, pp. 3253–3318, 2015.
- T. Fritz, T. Gonda, and P. Perrone, “De finetti’s theorem in categorical probability,” arXiv preprint arXiv:2105.02639, 2021.
- S. Moss and P. Perrone, “A category-theoretic proof of the ergodic decomposition theorem,” Ergodic Theory and Dynamical Systems, vol. 43, no. 12, p. 4166–4192, 2023.
- H.-O. Georgii, Gibbs Measures and Phase Transitions. Berlin, New York: De Gruyter, 2011.
- P. Alquier, “Approximate Bayesian Inference,” Entropy, vol. 22, p. 1272, Nov. 2020.
- L. Onsager, “Crystal statistics. i. a two-dimensional model with an order-disorder transition,” Phys. Rev., vol. 65, pp. 117–149, Feb 1944.
- S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, and A. Vichi, “Solving the 3d ising model with the conformal bootstrap,” Phys. Rev. D, vol. 86, p. 025022, Jul 2012.
- D. Simson, “Linear representations of partially ordered sets and vector space categories,” 1993.
- C.-S. Hu, “A brief note for sheaf structures on posets,” 2020.
- K. Yanagawa, “Sheaves on finite posets and modules over normal semigroup rings,” Journal of Pure and Applied Algebra, vol. 161, no. 3, pp. 341–366, 2001.
- A. Brown and O. Draganov, “Computing minimal injective resolutions of sheaves on finite posets,” 2021.
- E. Spiegel and C. J. O’Donnell, Incidence Algebras. CRC Press, 1997.
- M. Giry, “A categorical approach to probability theory,” in Categorical aspects of topology and analysis, vol. 915 of Lecture notes in Mathematics, pp. 68–85, Springer, 1982.
- E. W. Lawvere, “The category of probabilistic mappings,” 1962. Link.
- Springer Science & Business Media, 2013.
- World Scientific, 2022.
- Boston, MA: Springer US, 2009.
- A. De Martino and D. De Martino, “An introduction to the maximum entropy approach and its application to inference problems in biology,” Heliyon, vol. 4, no. 4, p. e00596, 2018.
- J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1988.
- S. L. Lauritzen, Graphical Models. Oxford Science Publications, 1996.
- E. Sudderth and J. Pacheco, “Probabilistic graphical models.” Brown University CSCI 2950-P, Spring 2013, Lecture 12: Gaussian Belief Propagation, State Space Models and Kalman Filters, Guest Kalman Filter Lecture by Jason Pacheco, 2013.
- T. Morita, “Cluster variation method of cooperative phenomena and its generalization i,” Journal of the Physical Society of Japan, vol. 12, no. 7, pp. 753–755, 1957.
- O. Peltre, “A homological approach to belief propagation and bethe approximations,” in Geometric Science of Information (F. Nielsen and F. Barbaresco, eds.), (Cham), pp. 218–227, Springer International Publishing, 2019.
- T. P. Speed, “A note on nearest-neighbour gibbs and markov probabilities,” Sankhyā: The Indian Journal of Statistics, Series A, 1979.
- G.-C. Rota, “On the foundations of combinatorial theory I. Theory of Möbius functions,” Probability theory and related fields, vol. 2, no. 4, pp. 340–368, 1964.
- H. K. T., “On the amount of information,” Theory of Probability and its Applications, 1962.
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