Geometric structures of statistical physics, information geometry, and learning : SPIGL'20, Les Houches, France, July 27-31 / / Frédéric Barbaresco and Frank Nielsen (editors)

• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Descrizione fisica: 1 online resource (466 pages)
• Disciplina: 530.13
• Soggetto topico: Física estadística; Intel·ligència artificial; Statistical physics; Artificial intelligence
• Soggetto genere / forma: Congressos; Llibres electrònics
• Persona (resp. second.): NielsenFrank; BarbarescoFrédéric
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Contents -- Part I: Tribute to Jean-Marie Souriau Seminal Works -- Structure des Systèmes Dynamiques Jean-Marie Souriau's Book 50th Birthday -- 1 A Few Introductory Words -- 2 Introduction -- 3 Chapter I: Differential Geometry -- 4 Chapter II: Symplectic Geometry -- 5 Chapter III: Mechanics -- 6 Chapter IV: Statistical Mechanics -- 7 Chapter V: A Method of Quantization -- 8 Conclusions -- References -- Jean-Marie Souriau's Symplectic Model of Statistical Physics: Seminal Papers on Lie Groups Thermodynamics - Quod Erat Demonstrandum -- 1 Preamble -- 2 Jean-Marie Souriau Biography -- 3 1st Souriau Paper: "Statistical Mechanics, Lie Group and Cosmology - 1st Part: Symplectic Model of Statistical Mechanics" -- 3.1 Distribution Functions -- 3.2 Statistical States -- 3.3 Image of Measures -- 3.4 Tensorial Products of Measure -- 3.5 Entropy -- 3.6 Canonical Gibbs Ensemble -- 3.7 Gibbs Ensemble of a Dynamic Group -- 3.8 Broken Symmetries -- 3.9 Thermodynamic Applications -- 3.10 Relativistic Thermodynamics -- 3.11 What is a Thermodynamic Equilibrium? -- 3.12 Proof of the Theorem (12) -- 4 2nd Souriau Paper: "Symplectic Geometry and Mathematical Physics" -- 4.1 1 - Since 1788. The Mechanics are Symplectic -- 4.2 2 - Emmy Noether and Measurable Quantities -- 4.3 3 - Mass and Cosmology -- 4.4 7 - Thermodynamics and Lie Groups -- 4.5 8 - Why the Earth Turns -- 5 3rd Souriau Paper: "Classical Mechanics and Symplectic Geometry" -- 5.1 Statistical Mechanics (Chapter 3.2) -- 5.2 Galilean Relativity (Chapter 2.7 in Souriau Paper) -- References -- Part II: Lie Group Geometry and Diffeological Model of Statistical Physics and Information Geometry -- Souriau-Casimir Lie Groups Thermodynamics and Machine Learning -- 1 Preamble -- 2 Souriau Lie Groups Thermodynamics and Covariant Gibbs Density -- 2.1 Geometric Structure of Information. ; 2.2 Lie Groups Thermodynamics and Souriau-Koszul-Fisher Metric -- 2.3 Souriau Entropy and Souriau-Fisher-Koszul Metric Invariance and Covariant Souriau Gibbs Density -- 3 New Entropy Characterization as Generalized Casimir Invariant Function in Coadjoint Representation -- 3.1 Souriau Entropy as Generalized Casimir Invariant in Coadjoint Representation -- 3.2 Souriau Entropy Invariance in Coadjoint Representation -- 3.3 Algebraic Method for Construction of Casimir Invariant Functions in Coadjoint Representation -- 4 Souriau Gibbs Density for Classical Lie Groups -- 4.1 Gibbs Density for SU(1,1) Lie Groups and Poincaré Disk in Case of Null Cohomology -- 4.2 Gibbs Density for SE(2) Lie Groups in Case of Non-null Cohomology -- 5 Conclusion -- References -- An Exponential Family on the Upper Half Plane and Its Conjugate Prior -- 1 Introduction -- 1.1 G/H-Method -- 1.2 Poincaré Distribution -- 1.3 Conjugate Prior of Exponential Family -- 2 Main Theorem -- 2.1 Main Theorem -- 2.2 Proof of Proposition 4 -- References -- Wrapped Statistical Models on Manifolds: Motivations, The Case SE(n), and Generalization to Symmetric Spaces -- 1 Introduction -- 2 Some Classical Probability Densities on Manifolds -- 3 Some Important Characteristics of Statistical Models on Manifolds -- 3.1 Expression of the Density Functions -- 3.2 Moments -- 3.3 Invariances and Estimation -- 4 Probability Densities on SE(n) -- 4.1 Wrapped Models on SE(n) -- 4.2 Density Estimation on SE(n) -- 5 Towards a Generalization to Symmetric Spaces -- 6 Conclusion -- References -- Galilean Thermodynamics of Continua -- 1 Some Words of Introduction -- 2 Space-Time and Galileo's Group -- 3 Geometric Structure of Thermodynamics -- 4 Temperature Vector and Friction Tensor -- 5 Momentum Tensors and First Principle -- 6 Reversible Processes and Thermodynamical Potentials. ; 7 Dissipative Continuum and Second Principle -- References -- Nonparametric Estimations and the Diffeological Fisher Metric -- 1 Introduction -- 2 Diffeological Fisher Metric, Diffeological Fisher Distance and Probabilistic Morphisms -- 3 Diffeological Cramér-Rao Inequality -- 4 Diffeological Hausdorff-Jeffrey Measure -- 5 Conclusion and Outlook -- References -- Part III: Advanced Geometrical Models of Statistical Manifolds in Information Geometry -- Information Geometry and Integrable Hamiltonian Systems -- 1 Introduction -- 1.1 The Toda Lattice and the Flaschka Transform -- 1.2 The Peakons System -- 1.3 Information Geometry, Toda System and Peakon System -- 2 Jacobi Flows and String Equation -- 2.1 Stieltjes Theorem -- 2.2 Hamburger Theorems and Stieltjes Integral -- 2.3 Discrete String -- 3 Finite Information Geometry -- 4 Conclusions and Perspectives -- References -- Relevant Differential Topology in Statistical Manifolds -- 1 Prologue -- 2 Intoduction -- 3 Basic Definitions -- 3.1 The Canonical Koszul Class of a Symmetric Gauge Structure -- 3.2 The Canonical Koszul Class of a Gauge Structure -- 3.3 Gauge Extensions of Gauge Dynamics -- 3.4 Transverse Statistical Structures -- 4 Reductions of Homogeneous Statistical Models -- 4.1 Canonical Projective Systems of Affinely Foliated H-Homogeneous Manifolds -- 4.2 Projective Sequence of Homogeneous Affinely Foliated Manifolds -- 4.3 Relative Invariant Subordinate Foliations -- 4.4 Subordinate Foliations and Topology of < -- H, M> -- -- 4.5 Metric Rigidity of FE() -- 5 The Case of Fisher Information -- 5.1 -equivalence -- 6 Relevant Foliations in Statistical Manifolds -- 6.1 Statistical Manifolds -- 6.2 Gauge Differential Operators -- 6.3 Relevant Constructions in Gauge Structures -- 6.4 Relevant Foliations in Positive Statistical Manifols -- 6.5 Symplectic Statistical Foliations. ; 6.6 Almost Hermitian Foliations in Statistical Manifolds -- 6.7 Riemannian Statistical Foliations -- 6.8 -Family of 4-Webs in Statistical Models of Measurable Sets -- References -- A Lecture About the Use of Orlicz Spaces in Information Geometry -- 1 Introduction -- 2 Orlicz Spaces -- 3 Calculus of the Gaussian Space -- 4 Exponential Statistical Bundle -- 5 Gaussian Orlicz-Sobolev Spaces -- 6 Selected Bibliography -- References -- Quasiconvex Jensen Divergences and Quasiconvex Bregman Divergences -- 1 Introduction, Motivation, and Contributions -- 2 Divergences Based on Inequality Gaps of Quasiconvex or Quasiconcave Generators -- 2.1 Quasiconvex and Quasiconcave Difference Dissimilarities -- 2.2 Relationship of Quasiconvex Difference Distances with Jensen Difference Distances -- 2.3 Quasiconvex Difference Distances from the Viewpoint of Comparative Convexity -- 3 Bregman Divergences for Quasiconvex Generators -- 3.1 Quasiconvex Bregman Divergences as Limit Cases of Quasiconvex Jensen Divergences -- 3.2 The -averaged Quasiconvex Bregman Divergence -- 3.3 Multivariate Quasiconvex Generators Q -- 3.4 Quasiconvex Bregman Divergences as Limit Cases of Power Mean Bregman Divergences -- 3.5 Some Illustrating Examples of Quasiconvex Bregman Divergences -- 4 Statistical Divergences, Parametric Families of Distributions and Equivalent Parameter Divergences -- 5 Conclusion and Perspectives -- 6 Calculations Using a Computer Algebra System -- References -- Part IV: Geometric Structures of Mechanics, Thermodynamics and Inference for Learning -- Dirac Structures and Variational Formulation of Thermodynamics for Open Systems -- 1 Fundamentals of Open Systems -- 1.1 Stueckelberg's Formulation of Nonequilibrium Thermodynamics -- 1.2 An Illustrative Example of Open Systems -- 2 A Variational Formulation for Open Systems. ; 2.1 Fundamental Setting for Open Nonequilibrium Thermodynamics -- 2.2 A Lagrangian Variational Formulation for Open Systems -- 3 Dirac Formulation for Time-Dependent Nonholonomic Systems of Thermodynamic Type -- 3.1 Time-Dependent Constraints of Thermodynamic Type -- 3.2 Dirac Structures on Covariant Pontryagin Bundles -- 3.3 Dirac Dynamical Systems on the Covariant Pontryagin Bundle -- 3.4 The Lagrange-d'Alembert-Pontryagin Principle on the Covariant Pontryagin Bundle -- 4 Dirac Formulation for Open Thermodynamic Systems -- 4.1 Application to the Piston-Cylinder System with External Ports -- 4.2 Dirac Dynamical Systems on the Covariant Pontryagin Bundle -- References -- The Geometry of Some Thermodynamic Systems -- 1 Introduction -- 2 Contact Geometry -- 2.1 The Jacobi Structure of a Contact Manifold -- 2.2 Hamiltonian and Evolution Vector Fields -- 3 The Lagrangian Formalism -- 3.1 The Geometric Setting -- 3.2 Generalized Chetaev Principle -- 4 The Evolution Vector Field and Simple Mechanical Systems with Friction -- 4.1 About the First and Second Laws of Thermodynamics -- 4.2 Examples -- 5 Composed Thermodynamic Systems Without Friction -- 6 Geometric Integration of Thermodynamic Systems -- 6.1 Simple Thermodynamic Systems with Friction -- 6.2 Composed Thermodynamic Systems -- 6.3 ``Variational Integration'' of the Evolution Vector Field -- 7 Conclusions and Future Work -- References -- Learning Physics from Data: A Thermodynamic Interpretation -- 1 Introduction -- 2 Pattern Recognition in Statistical Physics and Thermodynamics -- 2.1 Reduction and Pattern Recognition -- 2.2 Reducing Dynamics, Thermodynamics -- 2.3 Reduced Dynamics -- 3 Pattern Recognition in Machine Learning -- 3.1 General Scheme -- 3.2 Reduced Manifold Recognition by POD -- 3.3 Reduced Vector Field -- 4 Illustration on Learning from Particle Dynamics. ; 4.1 Smoothed Particle Hydrodynamics.
• Titolo autorizzato: Geometric structures of statistical physics, information geometry, and learning
• ISBN: 3-030-77957-2
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466400503316
• Lo trovi qui: Univ. di Salerno
• Serie: Springer Proceedings in Mathematics and Statistics