دانلود کتاب Geometric Science of Information: 5th International Conference, GSI 2021, Paris, France, July 21–23, 2021, Proceedings (Lecture Notes in Computer Science, 12829)

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Preface
Organization
Abstracts of Keynotes
Structures of Poisson Geometry: Old and New
Exploring Quantum Statistics for Machine Learning
Some Insights on Statistical Divergences and Choice of Models
Gaussian States from a Symplectic Geometry Point of View
The Primary Visual Cortex as a Cartan Engine
Use and Abuse of “Digital Information” in Life Sciences, is Geometry of Information a Way Out?
Contents
Probability and Statistics on Riemannian Manifolds
From Bayesian Inference to MCMC and Convex Optimisation in Hadamard Manifolds
1 MAP Versus MMS
2 Are They Equal?
3 Metropolis-Hastings
4 The Empirical Barycentre
5 Riemannian Gradient Descent
A Symmetric Hadamard Spaces
A.1 Hadamard Manifolds
A.2 Symmetric Spaces
References
Finite Sample Smeariness on Spheres
1 Introduction
2 Finite Sample Smeariness on Spheres
3 Why is Finite Sample Smeariness Called Finite Sample Smeariness?
4 Correcting for Finite Sample Smeariness in Statistical Testing
5 Finite Sample Smeariness Universality
References
Gaussian Distributions on Riemannian Symmetric Spaces in the Large N Limit
1 Gaussian Distributions on Riemannian Symmetric Spaces
2 Partition Functions at Finite N and Random Matrices
3 Partition Functions in the Large N Limit and the Saddle-Point Equation
References
Smeariness Begets Finite Sample Smeariness
1 Introduction
2 Assumptions, Notation and Definitions
3 Smeariness Begets Finite Sample Smeariness
4 Directional Smeariness
5 Simulations
References
Online Learning of Riemannian Hidden Markov Models in Homogeneous Hadamard Spaces
1 Introduction
2 Hidden Markov Chains
3 Online Estimation of Hidden Markov Models
4 Computational Experiment
5 Conclusion
References
Sub-Riemannian Geometry and Neuromathematics
Submanifolds of Fixed Degree in Graded Manifolds for Perceptual Completion
1 Introduction
2 A Subriemannian Model of the Visual Cortex
2.1 Orientation and Curvature Selectivity
3 Graded Structures
3.1 Regular Submanifolds
3.2 Admissible Variations
3.3 Variation for Submanifolds
4 Application to Visual Perception
4.1 Implementation and Results
References
An Auditory Cortex Model for Sound Processing
1 Introduction
2 The Contact Space Approach in V1
3 The Model of A1
3.1 The Lift Procedure
3.2 Associated Interaction Kernel
3.3 Processing of the Lifted Signal
3.4 Algorithm Pipeline
4 Numerical Implementation
5 Denoising Experiments
References
Conformal Model of Hypercolumns in V1 Cortex and the Möbius Group. Application to the Visual Stability Problem
1 Introduction
2 Riemannian Spinor Model of Conformal Sphere
3 Conformal Spherical Model of Hypercolumns
3.1 Relation with Symplectic Sarti-Citti-Petitot Model
3.2 Principle of Invariancy
4 The Cental Projection and the Shift of Retina Images After Post-saccade Remappling
4.1 The Central Projection onto the Retina
4.2 Möbius Projective Model of Conformal Sphere
4.3 Remapping as a Conformal Transformation
References
Extremal Controls for the Duits Car
1 Introduction
2 Problem Formulation
3 Pontryagin Maximum Principle
3.1 The Case h1<0.
3.2 The Case h1>0
4 Extremal Controls and Trajectories
References
Multi-shape Registration with Constrained Deformations
1 Introduction
2 Background
2.1 Structured Large Deformations
2.2 Multi-shape Registration
3 Modular Multi-shape Registration Problem
4 Numerical Results
5 Conclusion
References
Shapes Spaces
Geodesics and Curvature of the Quotient-Affine Metrics on Full-Rank Correlation Matrices
1 Introduction
2 Quotient-Affine Metrics
2.1 The Quotient Manifold of Full-Rank Correlation Matrices
2.2 The Affine-Invariant Metrics and the Quotient-Affine Metrics
3 Fundamental Riemannian Operations
3.1 Vertical and Horizontal Distributions and Projections
3.2 Horizontal Lift and Metric
3.3 Geodesics
3.4 Levi-Civita Connection and Sectional Curvature
4 Illustration in Dimension 2
5 Conclusion
A Proof of the Sectional Curvature
References
Parallel Transport on Kendall Shape Spaces
1 Introduction
2 The Quotient Structure of Kendall Shape Spaces
2.1 The Pre-shape Space
2.2 The Shape Space
2.3 The Quotient Metric
2.4 Implementation in geomstats
2.5 Parallel Transport in the Shape Space
3 The Pole Ladder Algorithm
3.1 Description
3.2 Properties
3.3 Complexity
4 Numerical Simulations and Results
5 Conclusion and Future Work
References
Diffusion Means and Heat Kernel on Manifolds
1 Introduction
2 Basic Concepts and Definitions
3 Examples of Known Heat Kernels
4 Application to Smeariness in Directional Data
References
A Reduced Parallel Transport Equation on Lie Groups with a Left-Invariant Metric
1 Introduction
2 Notations
3 Left-Invariant Metric and Connection
4 Parallel Transport
4.1 Geodesic Equation
4.2 Reduced Parallel Transport Equation
4.3 Application
References
Currents and K-functions for Fiber Point Processes
1 Introduction
2 Fibers as Currents
3 The K-function
3.1 Statistical Setup
3.2 K-function for Fibers
4 Experiments
4.1 Generated Data Sets
4.2 Application to Myelin Sheaths
References
Geometry of Quantum States
Q -Information Geometry of Systems
1 Introduction
2 Q -Information Geometry
3 Application
4 Conclusions
References
Group Actions and Monotone Metric Tensors: The Qubit Case
1 Introduction
2 Group Actions and Monotone Metrics for the Qubit
2.1 First Case (A=0)
2.2 Second Case (A>0)
2.3 Third Case (A<0)
3 Conclusions
References
Quantum Jensen-Shannon Divergences Between Infinite-Dimensional Positive Definite Operators
1 Introduction
2 Finite-Dimensional Setting
3 The Infinite-Dimensional Setting
4 The RKHS Setting
References
Towards a Geometrization of Quantum Complexity and Chaos
1 Introduction
2 Time Evolved QGT and Spreading of Local Correlations
3 The Phase Diagram Away from Equilibrium
3.1 Time Evolution
3.2 Equilibration
3.3 Application to the Cluster-XY Model Phase Diagram
4 Conclusions and Outlook
References
Hunt\'s Colorimetric Effect from a Quantum Measurement Viewpoint
1 Introduction
2 States and Effects in Colorimetry
3 Colorimetric Attributes of Color Effects
4 Hunt\'s Colorimetric Effect
5 Discussion and Future Perspectives
References
Geometric and Structure Preserving Discretizations
The Herglotz Principle and Vakonomic Dynamics
1 Introduction
1.1 The Herglotz Variational Principle, Revisited
1.2 An Alternative Formulation of Herglotz Variational Principle
1.3 Vakonomic Constraints
1.4 Applications to Control
References
Structure-Preserving Discretization of a Coupled Heat-Wave System, as Interconnected Port-Hamiltonian Systems
1 Introduction and Main Results
1.1 A Simplified and Linearised Fluid-Structure Model
1.2 Main Contributions and Organisation of the Paper
2 Port-Hamiltonian Formalism
2.1 The Fluid Model
2.2 The Structure Model
2.3 The Coupled System
3 The Partitioned Finite Element Method
3.1 For the Fluid
3.2 For the Structure
3.3 Coupling by Gyrator Interconnection
4 Numerical Simulations
5 Conclusion
References
Examples of Symbolic and Numerical Computation in Poisson Geometry
1 Introduction
2 Installation and Syntax
3 Examples of Our Symbolic and Numerical Methods
3.1 Poisson Brackets
3.2 Hamiltonian Vector Fields
3.3 Modular Vector Fields
3.4 Poisson Bivector Fields with Prescribed Casimirs
3.5 Classification of Lie–Poisson Bivector Fields on R3
References
New Directions for Contact Integrators
1 Introduction
2 Contact Integrators: Theory
2.1 Contact Variational Integrators (CVI)
2.2 Contact Hamiltonian Integrators (CHI)
3 Contact Integrators: Applications
3.1 Perturbed Kepler Problem
3.2 Contact Oscillator with Quadratic Action
4 Conclusions
References
Information Geometry in Physics
Space-Time Thermo-Mechanics for a Material Continuum
1 Introduction
2 Framework for a Space-Time Formulation of Continuum Mechanics
3 Energy-Momentum Tensor
4 Space-Time Thermodynamics
5 A Space-Time Constitutive Model for Thermo-Elastic Solids
6 Formulation of the Space-Time Problem for Isotropic Thermo-Elastic Transformations
7 Space-Time Finite Element Computation
References
Entropic Dynamics Yields Reciprocal Relations
1 Introduction
2 Background - Exponential Family
3 Entropic Dynamics on the Exponential Family Manifold
3.1 Obtaining the Transition Probability
3.2 Entropic Dynamics as a Diffusion Process
4 Reciprocity
5 Conclusion and Perspectives
References
Lie Group Machine Learning
Gibbs States on Symplectic Manifolds with Symmetries
1 Introduction
2 Gibbs States Built with the Moment Map of the Hamiltonian Action of a Lie Group
3 Examples of Gibbs States on Symplectic Manifolds with Symmetries
3.1 Three-Dimensional Real Oriented Vector Spaces with a Scalar Product
3.2 Gibbs States on Two-Dimensional Spheres
3.3 Gibbs States on Pseudo-spheres and Other Coadjoint Orbits
4 Final Comments
References
Gaussian Distributions on the Space of Symmetric Positive Definite Matrices from Souriau’s Gibbs State for Siegel Domains by Coadjoint Orbit and Moment Map
Abstract
1 Statistics on Lie Groups Based on Souriau Model of Gibbs Density
2 Gauss Density on Poincaré Unit Disk
3 Gauss Density on Siegel Unit Disk
4 Gauss Density on Siegel Upper Half Plane
References
On Gaussian Group Convex Models
1 Introduction
2 Structure of the Convex Parameter Set
3 Existence Condition and an Explicit Expression of MLE
References
Exponential-Wrapped Distributions on SL(2,C) and the Möbius Group
1 Introduction
2 The Group SL(2,C) and the Möbius Group
3 Exponential-Wrapped Distributions
4 There Are No Group-Isotropic Probability Distributions
5 Conclusion
References
Information Geometry and Hamiltonian Systems on Lie Groups
1 Introduction
2 Fisher-Rao Semi-definite Metric and Amari-Chentsov Cubic Tensor for the Transformation Models
3 Fisher-Rao Geodesic Flow and Euler-Poincaré Equation on Compact Semi-simple Lie Algebra
References
Geometric and Symplectic Methods for Hydrodynamical Models
Multisymplectic Variational Integrators for Fluid Models with Constraints
1 Introduction
2 Barotropic and Incompressible Fluids
2.1 Barotropic Fluids
2.2 Incompressible Fluid Models
3 2D Discrete Barotropic and Incompressible Fluid Models
3.1 Multisymplectic Discretizations
3.2 Discrete Lagrangian and Discrete Hamilton Principle
3.3 Discrete Multisymplectic Form Formula and Discrete Noether Theorem
3.4 Numerical Simulations
4 3D Discrete Barotropic and Incompressible Fluid Models
5 Concluding Remarks
References
Metriplectic Integrators for Dissipative Fluids
1 Metriplectic Dynamics
2 Dissipative Hydrodynamics
3 Metriplectic Integrators
4 Summary
References
From Quantum Hydrodynamics to Koopman Wavefunctions I
1 Introduction
2 Geometry of Quantum Hydrodynamics
3 Koopman-van Hove Formulation of Classical Mechanics
3.1 Evolution of Koopman Wavefunctions
3.2 Strict Contact Transformations
3.3 Momentum Maps and the Classical Liouville Equation
4 Hydrodynamic Quantities and von Neumann Operators
References
From Quantum Hydrodynamics to Koopman Wavefunctions II
1 Introduction
2 The Quantum-Classical Wave Equation
3 Madelung Equations and Quantum-Classical Trajectories
4 Joint Quantum-Classical Distributions
5 A Simple Closure Model
References
Harmonic Analysis on Lie Groups
The Fisher Information of Curved Exponential Families and the Elegant Kagan Inequality
1 Introduction: Fisher Information
2 Exponential Families
3 The Kagan Inequality
References
Continuous Wavelet Transforms for Vector-Valued Functions
1 Introduction
2 General Results
3 The 3-Dimensional Similitude Group Case
References
Entropy Under Disintegrations
1 Introduction
2 Generalized Differential Entropy
2.1 Definition and AEP
2.2 Certainty, Positivity and Divergence
3 Chain Rule
3.1 Disintegration of Measures
3.2 Chain Rule Under Disintegrations
3.3 Locally Compact Topological Groups
References
Koszul Information Geometry, Liouville-Mineur Integrable Systems and Moser Isospectral Deformation Method for Hermitian Positive-Definite Matrices
Abstract
1 Liouville Complete Integrability and Information Geometry
2 The Isospectral Deformation Method for Hermitian Positive Definite Matrices
3 First Integrals Associated to Lax Pairs, Characteristic Polynomial and Souriau Algorithm
References
Flapping Wing Coupled Dynamics in Lie Group Setting
1 Introduction
2 Reduced Fluid-Solid Interaction Model
3 Viscosity Effects
4 Concluding Remarks
References
Statistical Manifold and Hessian Information Geometry
Canonical Foliations of Statistical Manifolds with Hyperbolic Compact Leaves
1 Introduction
2 Hessian Equation on Gauge Structures
3 A Canonical Representation of G
4 Application to Statistical Manifolds
5 Conclusion and Future Works
References
Open Problems in Global Analysis. Structured Foliations and the Information Geometry
1 Notation and Basic Notions
1.1 Differential Structure, Versus Parameterization
1.2 C Relation
1.3 Geometric Structure, Versus Topology of Fibrations
1.4 Geometric Structure Versus Global Analysis: Equations of (X, P(X))
1.5 Structured Foliations
1.6 Open Problems
2 Relevant Geometric Structures Which Impact the Information Geometry
3 The Aim of the Talk
3.1 Long-Awaited Characteristic Obstructions
References
Curvature Inequalities and Simons\' Type Formulas in Statistical Geometry
1 Introduction
2 Preliminaries
3 Algebraic and Curvature Inequalities
4 Simons\' Formula
References
Harmonicity of Conformally-Projectively Equivalent Statistical Manifolds and Conformal Statistical Submersions
1 Introduction
2 Preliminaries
3 Harmonic Maps of Conformally-Projectively Equivalent Statistical Manifolds
4 Conformal Statistical Submersion and Harmonic Map
4.1 Harmonicity of Conformal Statistical Submersion
References
Algorithms for Approximating Means of Semi-infinite Quasi-Toeplitz Matrices
1 Introduction
2 Means of Two Quasi-Toeplitz Matrices
3 Power Mean
4 Computing the Karcher Mean in QT
5 Conclusions
References
Geometric Mechanics
Archetypal Model of Entropy by Poisson Cohomology as Invariant Casimir Function in Coadjoint Representation and Geometric Fourier Heat Equation
Abstract
1 Introduction
2 Lie Groups Thermodynamics and Souriau-Fisher Metric
3 Entropy Characterization as Generalized Casimir Invariant Function in Coadjoint Representation
4 Koszul Poisson Cohomology and Entropy Characterization
References
Bridge Simulation and Metric Estimation on Lie Groups
1 Introduction
2 Notation and Background
3 Simulation of Bridges on Lie Groups
4 Importance Sampling and Metric Estimation on SO(3)
References
Constructing the Hamiltonian from the Behaviour of a Dynamical System by Proper Symplectic Decomposition
1 Introduction
2 The Modal Analysis as an Equivalence Problem
3 Intertwined Eigenproblems and Spectral Decomposition
4 Proper Symplectic Decomposition (PSD)
5 Convergence Properties of the Method
6 Numerical Application
7 Conclusions and Perspectives
References
Non-relativistic Limits of General Relativity
1 Introduction
2 The Non-relativistic Particle Action
3 A Non-relativistic Target Space Action
4 Equations of Motion
5 Discussion
References
Deformed Entropy, Cross-Entropy, and Relative Entropy
A Primer on Alpha-Information Theory with Application to Leakage in Secrecy Systems
1 Introduction
2 A Primer on -Information Theory
2.1 Notations
2.2 Definitions
2.3 Basic Properties
2.4 Data Processing Inequalities
3 Fano\'s Inequality Applied to a Side-Channel Attack
References
Schrödinger Encounters Fisher and Rao: A Survey
1 Optimal Transport and the Schrödinger Problem
2 The Bures-Wasserstein Distance and Gaussian Optimal Transport
3 The Non-commutative Fisher-Rao Space
4 Extended Entropy, the Heat Flow, and the Schrödinger Problem
References
Projections with Logarithmic Divergences
1 Introduction
2 Preliminaries
3 Dual Foliation and Projection
4 PCA with Logarithmic Divergences
References
Chernoff, Bhattacharyya, Rényi and Sharma-Mittal Divergence Analysis for Gaussian Stationary ARMA Processes
1 Introduction
2 Chernoff, Bhattacharyya, Rényi and Sharma-Mittal Divergences. Application to the Gaussian Case
2.1 Definitions
2.2 Expressions of the Divergences in the Gaussian Case
3 About w.s.s. Gaussian ARMA Processes
4 Analysis of the Increments of the Divergences
5 Illustrations
6 Conclusions and Perspectives
References
Transport Information Geometry
Wasserstein Statistics in One-Dimensional Location-Scale Models
1 Introduction
2 W-Estimator
3 W-Estimator in Location-Scale Model
4 Asymptotic Distribution of W-Estimator
References
Traditional and Accelerated Gradient Descent for Neural Architecture Search
1 Introduction
2 Our Algorithms
2.1 First Order Algorithm
2.2 Second Order Algorithm
2.3 NASGD and NASAGD
3 Experiments
4 Conclusions and Discussion
References
Recent Developments on the MTW Tensor
1 A Short Background on Optimal Transport
2 The Ma-Trudinger-Wang Theory
2.1 Insights from Convex Analysis and Pseudo-Riemannian Geometry
2.2 The Squared-Distance on a Riemannian Manifold
3 Kähler Geometry
4 Applications of the MTW Tensor to Hessian Geometry
4.1 Complex Surfaces Satisfying MTW(0)
References
Wasserstein Proximal of GANs
1 Introduction
2 Wasserstein Natural Proximal Optimization
2.1 Motivation and Illustration
2.2 Wasserstein Natural Gradient
2.3 Wasserstein Natural Proximal
3 Implicit Generative Models
4 Computational Methods
4.1 Affine Space Variational Approximation
4.2 Relaxation from 1-D
4.3 Algorithms
References
Statistics, Information and Topology
Information Cohomology of Classical Vector-Valued Observables
1 Introduction
2 Some Known Results About Information Cohomology
3 An Extended Model
3.1 Information Structure, Supports, and Reference Measures
3.2 Probability Laws and Probabilistic Functionals
4 Computation of 1-cocycles
4.1 A Formula for Gaussian Mixtures
4.2 Kernel Estimates and Main Result
References
Belief Propagation as Diffusion
1 Graphical Models
2 Marginal Consistency
3 Energy Conservation
4 Diffusions
References
Towards a Functorial Description of Quantum Relative Entropy
1 Introduction and Outline
2 The Categories of Hypotheses and Optimal Hypotheses
3 The Relative Entropy as a Functor
References
Frobenius Statistical Manifolds and Geometric Invariants
1 Introduction
2 Statistical Manifolds and Frobenius Manifolds
3 Statistical Gromov–Witten Invariants and Learning
References
Geometric Deep Learning
`3́9`42`\"̇613A``45`47`\"603ASU(1,1) Equivariant Neural Networks and Application to Robust Toeplitz Hermitian Positive Definite Matrix Classification
1 Introduction
1.1 Related Work and Contribution
1.2 Preliminaries
2 Equivariant Convolution Operator
2.1 Convolution on D
2.2 Kernel Modeling and Locality
2.3 Numerical Aspects
3 Application to THPD Matrix Classification
3.1 Classification Problem
3.2 Numerical Results
4 Conclusions and Further Work
References
Iterative SE(3)-Transformers
1 Introduction
2 Background
2.1 Protein Structure Prediction and Refinement
2.2 Equivariance and the SE(3)-Transformer
2.3 Equivariant Attention
3 Implementation of an Iterative SE(3)-Transformer
3.1 Gradient Flow in Single-Pass vs. Iterative SE(3)-Transformers
3.2 Hidden Representations Between Blocks
3.3 Weight Sharing
3.4 Gradient Descent
4 Experiments
A Network Architecture and Training Details
References
End-to-End Similarity Learning and Hierarchical Clustering for Unfixed Size Datasets
1 Introduction
2 Related Works
2.1 Hyperbolic Hierarchical Clustering
3 End-to-End Similarity Learning and HC
4 Experiments
5 Conclusion
References
Information Theory and the Embedding Problem for Riemannian Manifolds
1 The Embedding Problem for Riemannian Manifolds
2 The Role of Information Theory
3 Renormalization Group (RG) Flows
3.1 General Principles
3.2 An Example: Random Lipschitz Functions
3.3 Interpretation
4 Isometric Embedding of Finite Metric Spaces into Rq
References
cCorrGAN: Conditional Correlation GAN for Learning Empirical Conditional Distributions in the Elliptope
1 Introduction
2 Related Work
2.1 Sampling of Random Correlation Matrices
2.2 GANs in Finance
2.3 Geometry of the Elliptope in Machine Learning
3 Our Contributions
4 The Set of Correlation Matrices
4.1 The Elliptope and its Geometrical Properties
4.2 Financial Correlations: Stylized Facts
5 Learning Empirical Distributions in the Elliptope Using Generative Adversarial Networks
5.1 CorrGAN: Sampling Realistic Financial Correlation Matrices
5.2 Conditional CorrGAN: Learning Empirical Conditional Distributions in the Elliptope
6 Application: Monte Carlo Simulations of Risk-Based Portfolio Allocation Methods
References
Topological and Geometrical Structures in Neurosciences
Topological Model of Neural Information Networks
1 Motivation
1.1 Homology as Functional to Processing Stimuli
1.2 Neural Code and Homotopical Representations
1.3 Consciousness and Informational Complexity
2 Homotopical Models of Neural Information Networks
2.1 Gamma Spaces and Information Networks
2.2 Neural Codes and Coding Theory
2.3 Dynamics of Networks in a Categorical Setting
2.4 Combined Roles of Topology
3 Toward a Model of Qualia and Experience
3.1 Neural Correlates of Consciousness and the Role of Topology
References
On Information Links
1 Introduction
2 Information Functions - Definitions
3 Information k-Links
4 Negativity and Kirkwood Decomposition Inconsistency
References
Betti Curves of Rank One Symmetric Matrices
1 Introduction
2 Betti Curves of Symmetric Rank 1 Matrices
3 Application to Calcium Imaging Data in Zebrafish Larvae
4 Conclusion
References
Algebraic Homotopy Interleaving Distance
1 Introduction
2 Multiparameter Persistence
3 Homotopy and Derived Interleavings
4 Application: Explaining the Instability of the Graded-Betti Numbers of Persistence Modules
5 Future Directions of Work
References
A Python Hands-on Tutorial on Network and Topological Neuroscience
1 Introduction
1.1 Starting Point: The Adjacency Matrix
1.2 Topological Data Analysis
1.3 Discussion
References
Computational Information Geometry
Computing Statistical Divergences with Sigma Points
1 Introduction
2 Exponential Families and Kullback-Leibler Divergence
3 Sigma Points for the Kullback-Leibler Divergence
3.1 Assessing the Errors of Monte Carlo Estimators
4 Sigma Points for the q-divergences
References
Remarks on Laplacian of Graphical Models in Various Graphs
1 Introduction
2 Trees
3 Discussion and Non-tree Graphs
3.1 Non-tree Graphs
3.2 Eigenvalues of L*
3.3 Discussion
References
Classification in the Siegel Space for Vectorial Autoregressive Data
1 Complex Vectorial Autoregressive Gaussian Models
1.1 The Multidimensional Linear Autoregressive Model
1.2 Three Equivalent Representation Spaces
1.3 A Natural Metric Coming from Information Geometry
2 The Siegel Disk
2.1 The Space Definition
2.2 The Metric
2.3 The Isometry
2.4 The Riemannian Logarithm Map
2.5 The Riemannian Logarithm Map at Any Point
2.6 The Riemannian Exponential Map
2.7 The Riemannian Exponential Map at Any Point
2.8 The Geodesics
2.9 Sectional Curvature of the Siegel Space
2.10 The Symmetric Siegel Disk
3 Application to Stationary Signals Classification
References
Information Metrics for Phylogenetic Trees via Distributions of Discrete and Continuous Characters
1 Introduction
2 Background
2.1 Phylogenetic Trees
2.2 Substitution Models and Character Distributions
3 Information Metrics from Distributions of Discrete Characters
3.1 Definition
3.2 Properties
3.3 Comparison of Metrics Under 2-State and 4-State Models
4 Intrinsic Geometry for Fixed Tree Topology
References
Wald Space for Phylogenetic Trees
1 Introduction
2 Wald Space
3 Geodesics in Wald Space
4 Comparing Fréchet Means
References
A Necessary Condition for Semiparametric Efficiency of Experimental Designs
1 Introduction
2 The Model
2.1 Tangent Space of a Statistical Model
2.2 Score Operator
2.3 Efficiency Bound
3 Main Results
4 Examples
4.1 Model Without Information Loss
4.2 Parametric Models
4.3 Dichotomous Choice Contingent Valuation Experiment
References
Parametrisation Independence of the Natural Gradient in Overparametrised Systems
1 Introduction
2 Parametrisation (In)dependence of the Natural Gradient
2.1 Gradient in Non-singular Points, pSmooth(M)
2.2 Gradient in Singular Points, p -.25ex-.25ex-.25ex-.25exSmooth(M)
3 Conclusion
References
Properties of Nonlinear Diffusion Equations on Networks and Their Geometric Aspects
1 Introduction
2 Preliminaries
3 Nonlinear Diffusion Equations on Finite Graphs
4 Convergence Rate
5 Information Geometric Viewpoints
6 Concluding Remarks
References
Rényi Relative Entropy from Homogeneous Kullback-Leibler Divergence Lagrangian
1 Information Geometry on the Statistical Bundle
2 From Divergences on E()E() to Lagrangians on SE()
3 Kullback-Leibler Dissimilarity Functional
4 Re-parametrization Invariance of the Dissimilarity Action
5 Finsler Structure on the Extended Statistical Bundle
6 Re-parametrization Invariance Gives (1/)-Rényi Divergence
References
Statistical Bundle of the Transport Model
1 Introduction
2 ANOVA
3 Gradient Flow of the Transport Problem
References
Manifolds and Optimization
Efficient Quasi-Geodesics on the Stiefel Manifold
1 Introduction
2 The Stiefel Manifold
3 Quasi-Geodesics on the Stiefel Manifold
3.1 Economy-Size Quasi-Geodesics
3.2 Short Economy-Size Quasi-Geodesics
4 Numerical Comparison
5 Conclusion and Outlook
References
Optimization of a Shape Metric Based on Information Theory Applied to Segmentation Fusion and Evaluation in Multimodal MRI for DIPG Tumor Analysis
1 Introduction
2 Proposition of a Criterion Based on Information Theory
2.1 Justification of the Criterion
2.2 Modelization of the Criterion in a Continuous Variational Setting
3 Multimodality Brain Tumors Segmentation Fusion and Evaluation
References
Metamorphic Image Registration Using a Semi-lagrangian Scheme
1 Introduction
2 Methods
3 Results and Conclusions
References
Geometry of the Symplectic Stiefel Manifold Endowed with the Euclidean Metric
1 Introduction
2 Geometry of the Riemannian Submanifold Sp(2p,2n)
3 Application to Optimization
4 Numerical Experiments
References
Divergence Statistics
On f-divergences Between Cauchy Distributions
1 Introduction
2 Information Geometry of Location-Scale Families
3 f-divergences Between Univariate Cauchy Distributions
3.1 Revisiting the KLD Between Cauchy Densities
3.2 f-divergences Between Cauchy Distributions are Symmetric
3.3 Maximal Invariants (Proof of Proposition 1)
4 Asymmetric KLD Between Multivariate Cauchy Distributions
References
Transport Information Hessian Distances
1 Introduction
2 Review of Transport Information Hessian Metric
2.1 Wasserstein Space
2.2 Hessian Metrics in Wasserstein Space
3 Transport Information Hessian Distances
3.1 Formulations
3.2 Properties
4 Closed-Form Distances
References
Minimization with Respect to Divergences and Applications
1 Introduction
2 Projection Using Divergences
3 Mimisation of Couple Matchings
4 Applications
4.1 Guessing Problem
4.2 Task Partitioning
References
Optimal Transport with Some Directed Distances
1 Introduction
2 A Toolkit of Divergences Between Quantile Functions
2.1 Quantile Functions
2.2 Directed Distances—Basic Concept
2.3 Justification of Distance Properties
3 New Optimal Transport Problems
References
Robust Empirical Likelihood
1 Introduction
2 A Robust Version of the Empirical Likelihood Estimator
2.1 Robustness Property
2.2 Asymptotic Properties
References
Optimal Transport and Learning
Mind2Mind: Transfer Learning for GANs
1 Introduction
2 Preliminaries
3 Mind to Mind Algorithm
4 Theoretical Guarantee for Convergence
5 Evaluation
6 Discussion
References
Fast and Asymptotic Steering to a Steady State for Networks Flows
1 Introduction
2 Discrete Schrödinger Bridges
3 Optimal Steering to a Steady State
3.1 Minimizing the Relative Entropy Rate
3.2 Existence for the One-Step Schrödinger System
4 Reversibility
References
Geometry of Outdoor Virus Avoidance in Cities
1 Introduction
2 Covid Exposure During Outdoor Excursion in Cities
3 Path Selection During Lock-Downs
3.1 Ariadne String Protocol Description
3.2 Estimation of Exposure Reduction During Covid-19 Lock-Downs
4 Path Selection During Lock-Downs Periods
4.1 Path Selection Algorithm
4.2 The Geo-Routing Algorithm
4.3 Simulation of the Algorithms
5 Conclusions
References
A Particle-Evolving Method for Approximating the Optimal Transport Plan
1 Introduction
2 Constrained Entropy Transport as a Regularized Optimal Transport Problem
2.1 Optimal Transport Problem and Its Relaxation
2.2 Constrained Entropy Transport Problem and Its Properties
3 Wasserstein Gradient Flow Approach for Solving the Regularized Problem
3.1 Wasserstein Gradient Flow
3.2 Wasserstein Gradient Flow of Entropy Transport Functional
3.3 Relating the Wasserstein Gradient Flow to a Particle System
4 Algorithmic Development
5 Numerical Experiments
6 Conclusion
References
Geometric Structures in Thermodynamics and Statistical Physics
Schrödinger Problem for Lattice Gases: A Heuristic Point of View
1 Introduction
2 Schrödinger Problem for Lattice Gases (SPLG)
3 Optimality Conditions
4 Convexity Along Optimal Flow
4.1 Convexity Under the Generalized McCann Condition
References
A Variational Perspective on the Thermodynamics of Non-isothermal Reacting Open Systems
1 Variational Formulation in Nonequilibrium Thermodynamics
1.1 Variational Formulation in Mechanics
1.2 Variational Formulation for Isolated Thermodynamic Systems
1.3 Variational Formulation for Open Thermodynamics Systems
2 Open Reacting Systems
2.1 General Setting, Thermodynamic Forces and Displacements
2.2 Variational Setting
2.3 The First and Second Law of Thermodynamics
2.4 Entropy Production Associated to Chemical Reactions
References
On the Thermodynamic Interpretation of Deep Learning Systems
1 Introduction
2 Thermodynamics and Deep Learning
3 Experiments
4 Contact Hamiltonian Dynamics, Lie Groups Thermodynamics and SGD
5 Conclusions
References
Dirac Structures in Thermodynamics of Non-simple Systems
1 Variational Formulation of Non-simple Systems
1.1 Setting for Thermodynamics of Non-simple Systems
1.2 Variational Formulation of Non-simple Systems
2 Dirac Formulation of Non-simple Systems
2.1 Dirac Structures in Thermodynamics
2.2 Dirac Formulation for Thermodynamics of Non-simple Systems
2.3 Example of the Adiabatic Piston
References
Author Index