Information Geometry and Statistical Manifold

Suzuki, Mashbat

doi:10.48550/arxiv.1410.3369

Cited by 3 publications

(4 citation statements)

References 2 publications

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“…Indeed, information geometry [1][2][3], which started with the seminal paper by Rao [4] has emerged from studies of invariant geometrical structure involved in statistical inference. It defines a Riemannian metric together with dually coupled affine connections in a manifold of probability distributions.…”

Section: Introductionmentioning

confidence: 99%

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Thermodynamic geometry of Nambu–Jona Lasinio model

2020

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The formalism of Riemannian geometry is applied to study the phase transitions in Nambu -Jona Lasinio (NJL) model. Thermodynamic geometry reliably describes the phase diagram, both in the chiral limit and for finite quark masses. The comparison between the geometrical study of NJL model and of (2+1) Quantum Chromodynamics at high temperature and small baryon density shows a clear connection between chiral symmetry restoration/breaking and deconfinement/confinement regimes.

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Section: Introductionmentioning

confidence: 99%

“…These geometric structures play important roles not only in statistical inference but also in wider areas of information sciences, such as machine learning, signal processing, optimization, neuroscience, mathematics and, of course, physics [1][2][3].…”

Section: Introductionmentioning

confidence: 99%

Thermodynamic geometry of Nambu–Jona Lasinio model

2020

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“…Let P be a distribution on the domain X that signifies our belief of where the optimal candidate for l resides. We assume that P belongs to the statistical manifold P [45] which is a Riemannian manifold [40] of probability distributions. Any point P ∈ P is expressed in the coordinates θ ∈ R n .…”

Section: Notationmentioning

confidence: 99%

CoNES: Convex Natural Evolutionary Strategies

Veer,

Majumdar

2020

Preprint

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We present a novel algorithm -convex natural evolutionary strategies (CoNES) -for optimizing high-dimensional blackbox functions by leveraging tools from convex optimization and information geometry. CoNES is formulated as an efficiently-solvable convex program that adapts the evolutionary strategies (ES) gradient estimate to promote rapid convergence. The resulting algorithm is invariant to the parameterization of the belief distribution. Our numerical results demonstrate that CoNES vastly outperforms conventional blackbox optimization methods on a suite of functions used for benchmarking blackbox optimizers. Furthermore, CoNES demonstrates the ability to converge faster than conventional blackbox methods on a selection of OpenAI's MuJoCo reinforcement learning tasks for locomotion.

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“…A point on an n-dimensional statistical manifold, D (from here on, we will use the symbol D to denote a statistical manifold unless specifically mentioned otherwise), can be identified with a (smooth) probability distribution function on a measurable topological space Ω, denoted by P (x; θ) [48,4]. Here, each distribution function can be parametrized using n real variables (θ…”

Section: Statistical Manifolds: Mathematical Preliminariesmentioning

confidence: 99%

Dictionary Learning and Sparse Coding on Statistical Manifolds

Chakraborty¹,

Banerjee²,

Vemuri³

2018

Preprint

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In this paper, we propose a novel information theoretic framework for dictionary learning (DL) and sparse coding (SC) on a statistical manifold (the manifold of probability distributions). Unlike the traditional DL and SC framework, our new formulation does not explicitly incorporate any sparsity inducing norm in the cost function being optimized but yet yields sparse codes. Our algorithm approximates the data points on the statistical manifold (which are probability distributions) by the weighted Kullback-Leibeler center/mean (KL-center) of the dictionary atoms. The KL-center is defined as the minimizer of the maximum KL-divergence between itself and members of the set whose center is being sought. Further, we prove that the weighted KL-center is a sparse combination of the dictionary atoms. This result also holds for the case when the KL-divergence is replaced by the well known Hellinger distance. From an applications perspective, we present an extension of the aforementioned framework to the manifold of symmetric positive definite matrices (which can be identified with the manifold of zero mean gaussian distributions), Pn. We present experiments involving a variety of dictionary-based reconstruction and classification problems in Computer Vision. Performance of the proposed algorithm is demonstrated by comparing it to several state-ofthe-art methods in terms of reconstruction and classification accuracy as well as sparsity of the chosen representation.

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Information Geometry and Statistical Manifold

Abstract: We review basic notions in the field of information geometry such as Fisher metric on statistical manifold, α-connection and corresponding curvature following Amari's work [1,2,3]. We show application of information geometry to asymptotic statistical inference.

Cited by 3 publications

References 2 publications

Thermodynamic geometry of Nambu–Jona Lasinio model

Thermodynamic geometry of Nambu–Jona Lasinio model

CoNES: Convex Natural Evolutionary Strategies

Dictionary Learning and Sparse Coding on Statistical Manifolds

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