Information geometry and statistical manifold

Abdel-All, Nassar H.; Abd-Ellah, H. N.; Moustafa, H. M.

doi:10.1016/s0960-0779(02)00142-x

Cited by 14 publications

(8 citation statements)

References 12 publications

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“…The Fisher information metric, which provides the similarity measurement, can be used to define the metric on the Riemannian manifold. It can be computed as [37]:…”

Section: Text Distance Metrics With Statistical Manifold Learningmentioning

confidence: 99%

Latent Topic Text Representation Learning on Statistical Manifolds

Jiang

Li²,

Chen

et al. 2018

IEEE Trans. Neural Netw. Learning Syst.

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Abstract-The explosive growth of text data requires effective methods to represent and classify these texts. Many text learning methods have been proposed, like statistics-based methods, semantic similarity methods and deep learning methods. The statistics-based methods focus on comparing the sub-structure of text, which ignores the semantic similarity between different words. Semantic similarity methods learn a text representation by training word embedding and representing text as the average vector of all words. However, these methods cannot capture the topic diversity of words and texts clearly. Recently, deep learning methods such as CNNs and RNNs have been studied. However, the vanishing gradient problem and time complexity for parameter selection limit their applications. In this paper, we propose a novel and efficient text learning framework, named Latent Topic Text Representation Learning (LTTR). Our method aims to provide an effective text representation and text measurement with latent topics. With the assumption that words on the same topic follow a Gaussian distribution, texts are represented as a mixture of topics, i.e., a Gaussian mixture model. Our framework is able to effectively measure text distance to perform text categorization tasks by leveraging statistical manifolds. Experimental results on text representation and classification, and topic coherence demonstrate the effectiveness of the proposed method.

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“…The Fisher information metric, which provides the similarity measurement, can be used to define the metric on the Riemannian manifold. It can be computed as [37]:…”

Section: Text Distance Metrics With Statistical Manifold Learningmentioning

confidence: 99%

Latent Topic Text Representation Learning on Statistical Manifolds

Jiang

Li²,

Chen

et al. 2018

IEEE Trans. Neural Netw. Learning Syst.

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“…The vector space is identified as the tangent space of S at θ, denoted as T θ S. With this structure in place, we can bring the machinery of Riemannian geometry to bear on statistical problems. In particular, the operations of covariant differentiation can be defined to describe the various connections of interest using the one-one correspondence between the statistical parameter model and Riemannian manifold [71].…”

Section: The Metric and Integrated Fisher Information Distancementioning

confidence: 99%

“…Rigorously speaking, the definition of a geodesic as a stationary point of the distance integral on a smooth manifold S with affine connection ∇ means that the curve γ(t) is such that parallel transport along the curve preserves the tangent vector to the curve [76]. Using local coordinates on S, the geodesic equations are given by the Euler-Lagrange equations as [71]…”

Section: Geodesics and Exponential Mapmentioning

confidence: 99%

Information Fusion and Control in Hierarchical Systems

Zhang¹,

Chong²,

Pezeshki³

et al. 2013

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“…However, when the inverse temperature increases, the underlying parametric space suffers a series of geometric deformations. Fisher information is capable of bringing insights about this complex process, since it is directly related to the metric tensor of the parametric space, being the most important ingredient to the computation of geodesic distances [1,17]. Statistical geometrodynamics have been proposed as a theoretical information geometric framework suitable to characterize chaotic dynamical behavior of arbitrary complex systems on curved statistical manifolds [14,11].…”

Section: Introductionmentioning

confidence: 99%

Information-theoretic analysis of complex systems modeled by two dimensional pairwise Ising models

Levada

2021

Preprint

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Stochastic complex systems are composed by a large number of seemingly simple variables that exhibit non-linear interactions with each other, causing the emergence of complexity and non-deterministic dynamics in the edge between order and chaos. Hence, the evolution of these systems seems to be completely out of control, with unpredictable behaviors. In this paper, using information geometry as a mathematical approach to chaos and complexity, we investigate how information theory can be used to analyze the dynamics of pairwise Ising random fields along Markov Chain Monte Carlo simulations in which phase transitions are observed. Our experiments indicate that Fisher information regarding the inverse temperature parameter can bring important insights, since it signalizes changes in the global spatial dependence structure. Information-theoretic curves are built to show that, despite the random nature of the system, it is possible to identify an asymmetric pattern of evolution when the system moves towards different entropic states.

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Information geometry and statistical manifold

Cited by 14 publications

References 12 publications

Latent Topic Text Representation Learning on Statistical Manifolds

Latent Topic Text Representation Learning on Statistical Manifolds

Information Fusion and Control in Hierarchical Systems

Information-theoretic analysis of complex systems modeled by two dimensional pairwise Ising models

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