Identifying features invariant to certain transformations is a fundamental problem in the fields of signal processing and pattern recognition. The earliest method of estimation of statistical parameters is the method of least squares due to Mark off. Define the Riemann metric on   SPD n as follows. total Bregman This article focuses on an important piece of work of the world renowned We show that Bhattacharyya distances on members of the same statistical exponential family amount to calculate a Burbea-Rao divergence in disguise. The proof uses standard techniques of differential geometry but does not use the language of category theory. Then we consider the dual structure of this manifold and investigate the Kullback divergence. This paper mainly contributes to a classification of statistical Einstein manifolds, namely statistical manifolds at the same time are Einstein manifolds. The statements, opinions and data contained in the journal, © 1996-2020 MDPI (Basel, Switzerland) unless otherwise stated. In this paper, we generalize the notions of centroids (and barycenters) to the broad class of information-theoretic distortion measures called Bregman divergences. In this paper a brief relatively nontechnical account is given of some relevant ideas in differential geometry. Several techniques to calculate or approximate f -divergences in general cases and for special distributions such as Gaussian and Gaussian mixture are reviewed. Evaluating the performance of Bayesian classification in a high-dimensional random tensor is a fundamental problem, usually difficult and under-studied. All rights reserved. of the singular behavior of the $L^{2}$-Wasserstein geometry. line of research with respect to other approaches is briefly discussed. You can request the full-text of this book directly from the authors on ResearchGate. The statements, opinions and data contained in the journals are solely Request PDF | On Mar 1, 2016, Huafei Sun and others published An Elementary Introduction to Information Geometry | Find, read and cite all the research you need on ResearchGate Thus we get an efficient algorithm for computing the Bhattacharyya centroid of a set of parametric distributions belonging to the same exponential families, improving over former specialized methods found in the literature that were limited to univariate or “diagonal” multivariate Gaussians. contrast which arises between the binary-alphabet and larger-alphabet settings. Our results extend Relative Fisher Information and Natural Gradient for Learning Large Modular Models, Teleparallel Gravity as a Higher Gauge Theory, On the Geometry of Mixtures of Prescribed Distributions, Monte Carlo Information Geometry: The dually flat case, Computational Information Geometry for Binary Classification of High-Dimensional Random Tensors, Patch Matching with Polynomial Exponential Families and Projective Divergences, Guaranteed Bounds on Information-Theoretic Measures of Univariate Mixtures Using Piecewise Log-Sum-Exp Inequalities, Total Jensen divergences: Definition, properties and clustering, A Novel Approach to Canonical Divergences within Information Geometry, On Monotone Embedding in Information Geometry, Hypothesis Testing, Information Divergence and Computational Geometry, An Information-Geometric Characterization of Chernoff Information, On Conformal Divergences and Their Population Minimizers, On the Chi Square and Higher-Order Chi Distances for Approximating f-Divergences, The uniqueness of the Fisher metric as information metric, Cramer-Rao Lower Bound and Information Geometry, An Information Geometric Approach to Polynomial-time Interior-point Algorithms—Complexity Bound via Curvature Integral—, Statistical manifolds are statistical models, The Burbea-Rao and Bhattacharyya Centroids, A Study on Invariance of -Divergence and Its Application to Speech Recognition, Fundamentals of Tensor Calculus for Engineers with a Primer on Smooth Manifolds, On the divergences of 1-conformally flat statistical manifolds, Reference duality and representation duality in information geometry, Divergence Functions and Geometric Structures They Induce on a Manifold, Information geometry and its applications, Theory of information space: A differential-geometrical foundation of statistics, Information geometry of the power inverse Gaussian distribution, Geometric Modeling in Probability and Statistics, Geometrical Foundation of Asymptotic Inference, On metric divergences of probability measures, The Role of Differential Geometry in Statistical Theory, Differential-geometrical methods in statistics. In this note we prove that any smooth (C1 resp.) In this work, we consider two Signal to Noise Ratio (SNR)-based binary classification problems of interest. Since PEFs have computationally intractable normalization terms, we estimate PEFs with score matching, and consider a projective distance: the symmetrized $$\gamma$$-divergence. This textbook is a unified presentation of differential geometry and probability theory, and constitutes a text for a course directed at graduate or advanced undergraduate students interested in applications of differential geometry in probability and statistics. differential geometry; metric tensor; affine connection; metric compatibility; conjugate connections; dual metric-compatible parallel transport; information manifold; statistical manifold; curvature and flatness; dually flat manifolds; Hessian manifolds; exponential family; mixture family; statistical divergence; parameter divergence; separable divergence; Fisher–Rao distance; statistical invariance; Bayesian hypothesis testing; mixture clustering; α-embeddings; mixed parameterization; gauge freedom, Help us to further improve by taking part in this short 5 minute survey, Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants, Effects of Glass Transition and Structural Relaxation on Crystal Nucleation: Theoretical Description and Model Analysis, Thermophoretic Micron-Scale Devices: Practical Approach and Review, dual metric-compatible parallel transport. We describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some uses of these information manifolds in information sciences. Appendices. of the research is focused on parametric statistical models. A matrix information-geometric method was developed to detect the change-points of rigid body motions. discovery by C.R. minimizers --- total Bregman divergences are indeed not symmetric. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. As the $$\alpha$$-Hessian structure is dually flat for $$\alpha = \pm 1$$, the $$\mathcal {D}_\varPhi$$-divergence provides richer geometric structures (compared to Bregman divergence) to the manifold $$\mathfrak {M}$$ on which it is defined.

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