Geometric Structures of Statistical Physics, Information Geometry, and Learning

✍ Scribed by Frédéric Barbaresco; Frank Nielsen

• Tongue: English
• Leaves: 466
• Category: Library

✦ Table of Contents

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
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
4.2 Reduced Manifold
4.3 Reduced Vector Field
4.4 Prediction
5 Illustration on Learning Rigid Body Mechanics
6 Conclusion
References
Computational Dynamics of Reduced Coupled Multibody-Fluid System in Lie Group Setting
1 Introduction
2 Dynamics of Multibody System in Fluid Flow
3 Added Mass Effect via Boundary Element Method
4 Vorticity Effects
5 Conclusion
References
Material Modeling via Thermodynamics-Based Artificial Neural Networks
1 Introduction
2 Artificial Neural Networks for Constitutive Modeling
2.1 Artificial Neural Networks
2.2 Material Modeling via Artificial Neural Networks
3 Thermodynamics-Based Artificial Neural Networks
3.1 Thermodynamics Principles and Theoretical Framework
3.2 Architecture of TANNs
4 Application to History-Dependent Materials with Softening
4.1 Material Model and Data
4.2 Training
4.3 Predictions in Recall Mode
4.4 TANNs Versus Standard ANNs
5 Conclusions
References
Information Geometry and Quantum Fields
1 Introduction
2 Symmetries and the Fisher Metric
3 Moduli Space of Instantons
4 The 2D Ising Model
5 The Curvature Menagerie
6 Discussion
References
Part V: Hamiltonian Monte Carlo, HMC Sampling and Learning on Manifolds
Geometric Integration of Measure-Preserving Flows for Sampling
1 Introduction
2 Monte Carlo Methods from Measure-Preserving Diffusions
3 A Complete and Canonical Characterisation of Measure-Preserving Diffusion
4 A Complete Characterisation of Measure-Preserving Mechanics
5 Hamiltonian Monte Carlo
References
Bayesian Inference on Local Distributions of Functions and Multidimensional Curves with Spherical HMC Sampling
1 Introduction
2 The Space of Smooth Curves
2.1 Curve Representation and Shape Space
2.2 Geodesics in Shape Space, Exponential Map and Parallel Transport
3 Analyzing Warping Functions
4 The Spherical Gaussian Process
5 The Bayesian Model
5.1 Maximum Likelihood
5.2 Spherical Hamiltonian Monte Carlo
6 Numerical Examples
6.1 Planar Curves
6.2 Spatial Curves
7 Conclusion
References
Sampling and Statistical Physics via Symmetry
1 Introduction
2 Sampling via Symmetry
2.1 Background
2.2 The Lie Group Generated by a Probability Measure
2.3 The Positive Monoid of a Measure
2.4 Barker and Metropolis Samplers
2.5 Some Algebra
2.6 Higher-Order Samplers
2.7 Behavior of Higher-Order Samplers
2.8 Linear Objectives for Transition Matrices
2.9 Remarks on Sampling
3 Statistical Physics via Symmetry
3.1 The Gibbs Distribution
3.2 Statistical and Physical System Descriptions
3.3 Preliminary Algebra
3.4 A Scaling Result
3.5 A Geometry Result
3.6 The Effective Temperature
3.7 Product Systems, the Ideal Gas, and Implications for t
3.8 Elementary Examples and Applications
3.9 Remarks
4 Application to Anosov Systems
4.1 Overview
4.2 Background on Anosov Systems
4.3 The Cat Map
4.4 Markov Partitions and Greedy Refinements
4.5 The Geodesic Flow on a Surface of Constant Negative Curvature
4.6 Conclusion
References
A Practical Hands-on for Learning Graph Data Communities on Manifolds
1 Introduction
2 Riemannian Community Embedding (RComE)
3 Code Structure and Dependencies
4 Implementing Manifold Specific Metrics and Operations
4.1 Metric
4.2 Exponential and Logarithmic Maps
4.3 Gaussian Distributions on Hyperbolic Spaces
5 Tutorial on RComE
5.1 Riemannian Gradient Descent Algorithms
5.2 An Example of Embedding Hierarchical Data
5.3 Preparing Graph Data: Synthetic Graphs and Real World Data
5.4 The First Order Loss
5.5 The Second Order Loss: Context and Negative Sampling
5.6 Learn Communities by Estimating a GMM via Expectation-Maximisation Algorithm
5.7 Community-Aware Node Embeddings
5.8 Learning Communities Graphs Embeddings
5.9 Evaluation of Learned Embeddings
6 Conclusion
References