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In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences.
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R. N. Sen · 1944
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R. N. Sen · 1944
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Aleksandr Petrovich Norden · 1945
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Information and the accuracy attainable in the estimation of statistical parameters
Radhakrishna C. Rao · 1945
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An invariant form for the prior probability in estimation problems
Harold Jeffreys · 1946
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Statistical decision functions
Abraham Wald · 1949
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An introduction to the theory of higher order spaces I. the theory of Kawaguchi spaces
Michiaki Kawaguchi · 1960
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The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming
Lev M. Bregman · 1967
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Information-type measures of difference of probability distributions and indirect observation
Imre Csiszár · 1967
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Defining the curvature of a statistical problem (with applications to second order efficiency)
Bradley Efron et al · 1975
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A relationship between the second derivatives of a convex function and of its conjugate
Jean-Pierre Crouzeix · 1977
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The geometrical structure of the parameter space of the two-dimensional normal distribution
Yoshiharu Sato, Kazuaki Sugawa, and Michiaki Kawaguchi · 1979
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Theory of information spaces: A differential geometrical foundation of statistics
Shun-ichi Amari · 1980
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Statistical decision rules and optimal inference
Nikolai N. Chentsov · 1982
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The Fréchet distance between multivariate normal distributions
D. C. Dowson and Basil V. Landau · 1982
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Differential geometry of smooth families of probability distributions
Hiroshi Nagaoka and Shun-ichi Amari · 1982
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Second order efficiency of minimum contrast estimators in a curved exponential family
Shinto Eguchi · 1983
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A Riemannian geometry of the multivariate normal model
Lene Theil Skovgaard · 1984
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A differential geometric approach to statistical inference on the basis of contrast functionals
Shinto Eguchi et al · 1985
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The role of differential geometry in statistical theory
Ole E. Barndorff-Nielsen, David Roxbee Cox, and Nancy Reid · 1986
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An extended Čencov characterization of the information metric
L. Lorne Campbell · 1986
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On manifolds with an affine connection and the theory of general relativity
Elie Joseph Cartan · 1986
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Geometrization of statistical theory. ULDM Publications, 1987
Christopher Terence John Dodson, editor · 1987
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Statistical manifolds
Stefan L. Lauritzen · 1987
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Statistical manifolds of univariate elliptic distributions
Ann F. S. Mitchell · 1988
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The information matrix, skewness tensor and
Ann F. S. Mitchell · 1989
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Differential-geometrical methods in statistics
Shun-ichi Amari · 1990
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Information and the accuracy attainable in the estimation of statistical parameters
C. Radhakrishna Rao · 1992
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A geometry for detection theory
Anand Ganesh Dabak · 1993
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Methods of Information Geometry
Shun ichi Amari and Hiroshi Nagaoka · 1993
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Differential geometry and statistics
Michael Murray and John Rice · 1993
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On the divergences of
Takashi Kurose · 1994
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Affine differential geometry: Geometry of affine immersions
Katsumi Nomizu, Nomizu Katsumi, and Takeshi Sasaki · 1994
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Inducing features of random fields
Stephen Della Pietra, Vincent Della Pietra, and John Lafferty · 1997
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Geometrical Foundations of Asymptotic Inference
Robert E. Kass and Paul W. Vos · 1997
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About distances of discrete distributions satisfying the data processing theorem of information theory
María del Carmen Pardo and Igor Vajda · 1997
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Natural gradient works efficiently in learning
Shun-ichi Amari · 1998
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A characterization of monotone and regular divergences
JM Corcuera and Federica Giummolè · 1998
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Chance, love, and logic: Philosophical essays
Charles Sanders Peirce · 1998
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Approximate solutions to Markov decision processes
G. J. Gordon · 1999
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Dual differential geometry associated with kullback-leibler information on the gaussian distributions and its 2-parameter deformations
Shintaro Yoshizawa and Kunio Tanabe · 1999
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Selection and reinforcement learning for combinatorial optimization
Arnaud Berny · 2000
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Relative loss bounds for on-line density estimation with the exponential family of distributions
Katy S Azoury and Manfred K Warmuth · 2001
Cited alongside, same era.
Evolution strategies–a comprehensive introduction
Hans-Georg Beyer and Hans-Paul Schwefel · 2002
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Absolute parallelism geometry: Developments, applications and problems
MI Wanas · 2002
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Computing the Rao distance for Gamma distributions
Ferran Reverter and Josep M. Oller · 2003
Cited alongside, same era.
Information theory and statistics: A tutorial
Imre Csiszár and Paul C Shields · 2004
Cited alongside, same era.
Jensen-Shannon divergence and Hilbert space embedding
Bent Fuglede and Flemming Topsøe · 2004
Cited alongside, same era.
An information-geometric characterization of Chernoff information
Frank Nielsen · 2013
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Geometric Science of Information, volume 8085 of Lecture Notes in Computer Science. Springer, 2013
Frank Nielsen and Frédéric Barbaresco, editors · 2013
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On the chi square and higher-order chi distances for approximating
Frank Nielsen and Richard Nock · 2013
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Nonparametric information geometry
Giovanni Pistone · 2013
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New developments of information geometry
Shun-ichi Amari · 2014
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Geometric Modeling in Probability and Statistics
Ovidiu Calin and Constantin Udriste · 2014
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Clustering with Bregman divergences
Arindam Banerjee, Srujana Merugu, Inderjit S Dhillon, and Joydeep Ghosh · 2005
Cited alongside, same era.
Quantum information
Masahito Hayashi · 2006
Cited alongside, same era.
Statistical manifolds are statistical models
Hông Vân Lê · 2006
Cited alongside, same era.
Methods of Information Geometry
Shun-ichi Amari and Hiroshi Nagaoka · 2007
Cited alongside, same era.
The natural philosophy of Kazuo Kondo
Grenville J Croll · 2007
Cited alongside, same era.
An information geometric approach to polynomial-time interior-point algorithms: Complexity bound via curvature integral
Atsumi Ohara and Takashi Tsuchiya · 2007
Cited alongside, same era.
Information measures: the curious case of the binary alphabet
Jiantao Jiao, Thomas A Courtade, Albert No, Kartik Venkat, and Tsachy Weissman · 2014
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Geometric Theory of Information
Frank Nielsen · 2014
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Natural evolution strategies
Daan Wierstra, Tom Schaul, Tobias Glasmachers, Yi Sun, Jan Peters, and Jürgen Schmidhuber · 2014
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Divergence functions and geometric structures they induce on a manifold
Jun Zhang · 2014
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A novel approach to canonical divergences within information geometry
Nihat Ay and Shun-ichi Amari · 2015
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Teleparallel gravity as a higher gauge theory
John C. Baez and Derek K. Wise · 2015
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Convex optimization: Algorithms and complexity
Sébastien Bubeck et al · 2015
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Foundations of Information Geometry
Akio Fujiwara · 2015
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Mathematical foundations of infinite-dimensional statistical models, volume 40
Evarist Giné and Richard Nickl · 2015
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Information geometry of the Gaussian distribution in view of stochastic optimization
Luigi Malagò and Giovanni Pistone · 2015
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Geometric Science of Information, volume 9389 of Lecture Notes in Computer Science. Springer, 2015
Frank Nielsen and Frédéric Barbaresco, editors · 2015
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Total Jensen divergences: Definition, properties and clustering
Frank Nielsen and Richard Nock · 2015
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The information geometry of mirror descent
Garvesh Raskutti and Sayan Mukherjee · 2015
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On monotone embedding in information geometry
Jun Zhang · 2015
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Reference duality and representation duality in information geometry
Jun Zhang · 2015
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Information Geometry and Its Applications
Shun-ichi Amari · 2016
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Information Geometry and its Applications: On the Occasion of Shun-ichi Amari’s 80th Birthday, volume 252 of Springer Proceedings in Mathematics & Statistics
Nihat Ay, Paolo Gibilisco, and Frantisek Matús · 2016
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Differential geometry of curves and surfaces: revised and updated second edition
Manfredo P Do Carmo · 2016
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Patch matching with polynomial exponential families and projective divergences
Frank Nielsen and Richard Nock · 2016
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Guaranteed bounds on information-theoretic measures of univariate mixtures using piecewise log-sum-exp inequalities
Frank Nielsen and Ke Sun · 2016
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Guaranteed bounds on the Kullback–Leibler divergence of univariate mixtures
Frank Nielsen and Ke Sun · 2016
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On conformal divergences and their population minimizers
Richard Nock, Frank Nielsen, and Shun-ichi. Amari · 2016
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Fundamentals of tensor calculus for engineers with a primer on smooth manifolds, volume 230
Uwe Mühlich · 2017
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Geometric Science of Information, volume 10589 of Lecture Notes in Computer Science. Springer, 2017
Frank Nielsen and Frédéric Barbaresco, editors · 2017
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On Hölder projective divergences
Frank Nielsen, Ke Sun, and Stéphane Marchand-Maillet · 2017
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Relative Fisher information and natural gradient for learning large modular models
Ke Sun and Frank Nielsen · 2017
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The uniqueness of the Fisher metric as information metric
Hông Vân Lê · 2017
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Rho–tau embedding and gauge freedom in information geometry
Jan Naudts and Jun Zhang · 2018
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Geometric Structures of Information
Frank Nielsen · 2018
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What is… an information projection?
Frank Nielsen · 2018
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Monte Carlo information geometry: The dually flat case
Frank Nielsen and Gaëtan Hadjeres · 2018
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On the geometry of mixtures of prescribed distributions
Frank Nielsen and Richard Nock · 2018
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Computational information geometry for binary classification of high-dimensional random tensors
Gia-Thuy Pham, Rémy Boyer, and Frank Nielsen · 2018
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Noisy natural gradient as variational inference
Guodong Zhang, Shengyang Sun, David Duvenaud, and Roger Grosse · 2018
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Geometry and fixed-rate quantization in riemannian metric spaces induced by separable Bregman divergences
Erika Gomes-Gonçalves, Henryk Gzyl, and Frank Nielsen · 2019
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On geodesic triangles with right angles in a dually flat space
Frank Nielsen · 2019
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On the Jensen–Shannon symmetrization of distances relying on abstract means
Frank Nielsen · 2019
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Monte Carlo information-geometric structures
Frank Nielsen and Gaëtan Hadjeres · 2019
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Ruppeiner geometry, phase transitions, and the microstructure of charged AdS black holes
Shao-Wen Wei, Yu-Xiao Liu, and Robert B Mann · 2019
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On Voronoi diagrams on the information-geometric Cauchy manifolds
Frank Nielsen · 2020
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Frank Nielsen and Richard Nock · 2020
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