Nielsen F., Srinivasa Rao A.S.R., Rao C.R. (eds.) Geometry and Statistics

London: Academic Press, 2022. — 490 p.Foundations in classical geometry and analysis
Geometry, information, and complex bundlesComplex planes
Important implications of Liouville's theorem
Geometric analysis and Jordan curvesGeometric methods for sampling, optimization, inference, and adaptive agentsAccelerated optimization
Principle of geometric integration
Conservative flows and symplectic integrators
Rate-matching integrators for smooth optimization
Manifold and constrained optimization
Gradient flow as a high friction limit
Optimization on the space of probability measures
Hamiltonian-based accelerated sampling
Optimizing diffusion processes for sampling
Hamiltonian Monte Carlo
Statistical inference with kernel-based discrepancies
Topological methods for MMDs
Smooth measures and KSDs
The canonical Stein operator and Poincaré duality
Kernel Stein discrepancies and score matching
Information geometry of MMDs and natural gradient descent
Minimum Stein discrepancy estimators
Likelihood-free inference with generative models
Adaptive agents through active inference
Modeling adaptive decision-making
Behavior, agents, and environments
Decision-making in precise agents
The information geometry of decision-making
Realizing adaptive agents
The basic active inference algorithm
Sequential decision-making under uncertainty
World model learning as inference
Scaling active inferenceEquivalence relations and inference for sparse Markov modelsImproved modeling capabilities of sparse Markov models (SMMs)
Fitting SMMs and example applications
Model fitting based on a collapsed Gibbs sampler
Modeling wind speeds
Modeling a DNA sequence
Fitting SMM through regularization
Application to classifying viruses
Equivalence relations and the computation of distributions of pattern statistics for SMMs
Notation
Computing distributions in higher-order Markovian sequences
Specializing the computation to SMM
Application to spaced seed coverageInformation geometry
Symplectic theory of heat and information geometry
Preamble
Life and seminal work of Souriau on lie groups thermodynamics
From information geometry to lie groups thermodynamics
Symplectic structure of fisher metric and entropy as Casimir function in coadjoint representation
Symplectic Fisher Metric structures given by Souriau model
Entropy characterization as generalized Casimir invariant function in coadjoint representation and Poisson Cohomology
Koszul Poisson Cohomology and entropy characterization
Covariant maximum entropy density by Souriau model
Gauss density on Poincaré unit disk covariant with respect to SU(1,1) Lie group
Gauss density on Siegel unit disk covariant with respect to SU(N,N) Lie group
Gauss density on Siegel upper half planeFurther reading
A unifying framework for some directed distances in statistics
Divergences, statistical motivations, and connections to geometry
Basic requirements on divergences (directed distances)
Some statistical motivations
Incorporating density function zeros
Some motivations from probability theory
Divergences and geometry
Some incentives for extensions
phi-Divergences between other statistical objects
Some non-phi-divergences between probability distributions
Some non-phi-divergences between other statistical objects
The framework
Statistical functionals S and their dissimilarity
The divergences (directed distances) D
The reference measure λ
The divergence generator phi
The scaling and the aggregation functions m1, m2, and m3
m1(x) = m2(x) =: m(x), m3(x) = r(x)m(x) [0, ] for some (measurable) function r:XR satisfying r(x)]-,0[]0,[ for λ-...
m1(x) = m2(x):= 1, m3(x) = r(x) for some (measurable) function r:X[0,] satisfying r(x) ]0, [ for λ-a.a. xX
m1(x) = m2(x):= Sx(Q), m3(x) = r(x)Sx(Q) [0, ] for some (measurable) function r:XR satisfying r(x)]-,0[]0,[ for...
m1(x) = m2(x):= w(Sx(P), Sx(Q)), m3(x) = r(x)w(Sx(P), Sx(Q)) [0, [ for some (measurable) functions w:R(S(P))xR(...
m1(x)=Sx(P) and m2(x)=Sx(Q) with statistical functional SS, m3(x) 0
Auto-divergences
Connections with optimal transport and coupling
Aggregated/integrated divergences
Dependence expressing divergences
Bayesian contexts
Variational representations
Some further variantsThe analytic dually flat space of the mixture family of two prescribed distinct Cauchy distributions
Introduction and motivation
Differential-geometric structures induced by smooth convex functions
Hessian manifolds and Bregman manifolds
Bregman manifolds: Dually flat spaces
Some illustrating examples
Exponential family manifolds
Natural exponential family
Fisher–Rao manifold of the categorical distributions
Regular cone manifolds
Mixture family manifolds
Definition
The categorical distributions: A discrete mixture family
Information geometry of the mixture family of two distinct Cauchy distributions
Cauchy mixture family of order 1
An analytic example with closed-form dual potentialsAppendix. Symbolic computing notebook in MAXIMALocal measurements of nonlinear embeddings with information geometryα-Divergence and autonormalizing
α-Discrepancy of an embedding
Empirical α-discrepancy
Connections to existing methods
Neighborhood embeddings
Autoencoders
Conclusion and extensions
Conclusion and extensions
Appendices
Appendix A. Proof of Lemma 1
Appendix B. Proof of Proposition 1
Appendix C. Proof of Proposition 2
Appendix D. Proof of Theorem 1Advanced geometrical intuition
Parallel transport, a central tool in geometric statistics for computational anatomy: Application to cardiac m...Diffeomorphometry
Longitudinal models
Parallel transport for intersubject normalization
Chapter organization
Parallel transport with ladder methods
Numerical accuracy of Schild's and pole ladders
Elementary construction of Schild's ladder
Taylor expansion
Numerical scheme and convergence
Pole ladder
Infinitesimal schemes
A short overview of the LDDMM framework
Ladder methods with LDDMM
Validation
Application to cardiac motion modeling
The right ventricle and its diseases
Motion normalization with parallel transport
Interaction between shape and deformations: A scale problem
An intuitive rescaling of LDDMM parallel transport
Hypothesis
Criterion and estimation of λ
Results
Relationship between λ and VolED
Changing the metric to preserve relative volume changes
Model
Implementation
Geodesics
Results
Analysis of the normalized deformations
Geodesic and spline regression
Results
Hotelling tests on velocitiesAbbreviationsGeometry and mixture modelsFundamentals of modeling with mixtures
Mixtures and the fundamentals of geometry
Structure of article
Identification, singularities, and boundaries
Mixtures of finite distributions
Likelihood geometry
General geometric structures
Singular learning theory
Bayesian methods
Singularities and algebraic geometry
Singular learning and model selection
Nonstandard testing problems
DiscussionGaussian distributions on Riemannian symmetric spaces of nonpositive curvatureGaussian distributions and RMT
From Gauss to Shannon
The ``right´´ Gaussian
The normalizing factor Z(σ)
MLE and maximum entropy
Barycenter and covariance
Z(σ) from RMT
The asymptotic distribution
Duality: The Θ distributions
Gaussian distributions and Bayesian inference
MAP versus MMS
Bounding the distance
Computing the MMS
Metropolis-Hastings algorithm
The empirical barycenter
Proof of Proposition 13
Appendix A. Riemannian symmetric spaces
The noncompact case
The compact case
Example of Propositions A.1 and A.2
Appendix B. Convex optimization
Convex sets and functions
Second-order Taylor formula
Taylor with retractions
Riemannian gradient descent
Strictly convex case
Strongly convex case
Appendix C. Proofs for Section BMultilevel contours on bundles of complex planes*Infinitely many bundles of complex planes
Multilevel contours in a random environment
Behavior of X (zl(t), l) at (l 0)
Loss of spaces in bundle B()
Islands and holes in B()
Consequences of B()\l on multilevel contours
PDEs for the dynamics of lost space
Concluding remarksBack Cover