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GLOBAL INFORMATION GEOMETRY
, 2002
"... Abstract. Dually flat manifolds constitute fundamental mathematical objects of information geometry. This note establishes some facts on the global properties and topology of dually flat manifolds which, in particular, provide answers to questions and problems in global information geometry posed by ..."
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Abstract. Dually flat manifolds constitute fundamental mathematical objects of information geometry. This note establishes some facts on the global properties and topology of dually flat manifolds which, in particular, provide answers to questions and problems in global information geometry posed
Information Geometry on Hierarchy of Probability Distributions
, 2001
"... An exponential family or mixture family of probability distributions has a natural hierarchical structure. This paper gives an “orthogonal” decomposition of such a system based on information geometry. A typical example is the decomposition of stochastic dependency among a number of random variables ..."
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Cited by 106 (8 self)
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An exponential family or mixture family of probability distributions has a natural hierarchical structure. This paper gives an “orthogonal” decomposition of such a system based on information geometry. A typical example is the decomposition of stochastic dependency among a number of random
IN QUANTUM INFORMATION GEOMETRY
, 2002
"... Abstract. We investigate questions in quantum information geometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finitedimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually fl ..."
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Abstract. We investigate questions in quantum information geometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finitedimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually
Information Geometry and Phase Transitions
, 2004
"... The introduction of a metric onto the space of parameters in models in Statistical Mechanics and beyond gives an alternative perspective on their phase structure. In such a geometrization, the scalar curvature, R, plays a central role. A noninteracting model has a flat geometry (R = 0), while R dive ..."
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diverges at the critical point of an interacting one. Here, the information geometry is studied for a number of solvable statisticalmechanical models.
Information Geometry of Contrastive Divergence
"... Abstract The contrastive divergence(CD) method proposed by Hinton finds an approximate solution of the maximum likelihood of complex probability models. It is known empirically that the CD method gives a highquality estimation in a small computation time. In this paper, we give an intuitive explan ..."
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explanation about the reason why the CD method can approximate well by using the information geometry. We further propose an improved method that is consistent with the maximum likelihood (or MAP) estimation, while the CD method is biased in general.
The Information Geometry of the Spherical Model
, 2008
"... Motivated by previous observations that geometrizing statistical mechanics offers an interesting alternative to more standard approaches, we have recently calculated the curvature (the fundamental object in this approach) of the information geometry metric for the Ising model on an ensemble of plana ..."
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Cited by 3 (0 self)
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Motivated by previous observations that geometrizing statistical mechanics offers an interesting alternative to more standard approaches, we have recently calculated the curvature (the fundamental object in this approach) of the information geometry metric for the Ising model on an ensemble
Information geometries for microeconomic theories
, 2009
"... More than thirty years ago, Charnes, Cooper and Schinnar (1976) established an enlightening contact between economic production functions (epfs) — a cornerstone of neoclassical economics — and information theory, showing how a generalization of the CobbDouglas production function encodes homogeneo ..."
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homogeneous functions. As expected by Charnes et al., the contact turns out to be much broader: we show how information geometry as pioneered by Amari 1 and others underpins static and dynamic descriptions of microeconomic cornerstones. We show that the most popular epfs are fundamentally grounded in a very
Legendre transformation and information geometry
, 2010
"... We explain geometrically the Legendre transformation for astrictly convex function x ∈ X ↦ → F(x), by first “plotting” its graph, and then reinterpreting this graph as the intersection of its supporting halfspaces. A supporting halfspace is parameterized by a dual “slope ” parameter y = ∇F(x) ∈ Y ..."
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metric distances BF,F ∗(x,y) = F(x)+F ∗ (y)−x T y ≥ 0 ∀x ∈ X,y ∈ Y that plays the role of canonical divergences of flat spaces in information geometry. Properties of the Legendre transformation are finally briefly listed.
Information Geometry of the EM and em Algorithms for Neural Networks
 Neural Networks
, 1995
"... In order to realize an inputoutput relation given by noisecontaminated examples, it is effective to use a stochastic model of neural networks. A model network includes hidden units whose activation values are not specified nor observed. It is useful to estimate the hidden variables from the obs ..."
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Cited by 120 (9 self)
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by information geometry. The emalgorithm minimizes iteratively the KullbackLeibler divergence in the manifold of neural networks. These two algorithms are equivalent in most cases. The present paper gives a unified information geometrical framework for studying stochastic models of neural networks
INFORMATION GEOMETRY AND STATISTICAL INFERENCE
"... Abstract. Variance and Fisher information are ingredients of the CramerRao inequality. Fisher information is regarded as a Riemannian metric on a quantum statistical manifold and we choose monotonicity under coarse graining as the fundamental property. The quadratic cost functions are in a dual rel ..."
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relation with the Fisher information quantities and they reduce to the variance in the commuting case. The scalar curvature in a certain geometry might be interpreted as an uncertainty on a statistical manifold. Information geometry has a surprising application to the theory of geometric mean of matrices
Results 1  10
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