Gaussian Random Fields

Hristopulos, Dionissios T.

doi:10.1007/978-94-024-1918-4_6

Cited by 4 publications

(3 citation statements)

References 57 publications

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“…The objects of interest in this study are pairwise isotropic Gaussian-Markov random fields, that are collections of spatially dependent variables organized in the vertices of a graph, whose set of observable states is continuous [12]. The degree of interaction between neighboring variables is quantified by a coupling parameter, also known as the inverse temperature.…”

Section: Introductionmentioning

confidence: 99%

On the Kullback-Leibler divergence between pairwise isotropic Gaussian-Markov random fields

Levada¹

2022

Preprint

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The Kullback-Leibler divergence or relative entropy is an information-theoretic measure between statistical models that play an important role in measuring a distance between random variables. In the study of complex systems, random fields are mathematical structures that models the interaction between these variables by means of an inverse temperature parameter, responsible for controlling the spatial dependence structure along the field. In this paper, we derive closed-form expressions for the Kullback-Leibler divergence between two pairwise isotropic Gaussian-Markov random fields in both univariate and multivariate cases. The proposed equation allows the development of novel similarity measures in image processing and machine learning applications, such as image denoising and unsupervised metric learning.

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Section: Introductionmentioning

confidence: 99%

On the Kullback-Leibler divergence between pairwise isotropic Gaussian-Markov random fields

Levada¹

2022

Preprint

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“…A relevant aspect concerns the prediction of phase transitions in a quantitative way by means of an objective mathematical criterion [7,8]. In this paper, we compute intrinsic geometric properties from the underlying parametric space of random fields composed by Gaussian variables [9] and study how these quantities change along phase transitions.…”

Section: Introductionmentioning

confidence: 99%

The curvature effect in Gaussian random fields

Levada¹

2022

J. Phys. Complex.

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Random field models are mathematical structures used in the study of stochastic complex systems. In this paper, we compute the shape operator of Gaussian random field manifolds using the first and second fundamental forms (Fisher information matrices). Using Markov chain Monte Carlo techniques, we simulate the dynamics of these random fields and compute the Gaussian, mean and principal curvatures of the parametric space, analyzing how these quantities change along dynamics exhibiting phase transitions. During the simulations, we have observed an unexpected phenomenon that we called the curvature effect, which indicates that a highly asymmetric geometric deformation happens in the underlying parametric space when there are significant increase/decrease in the system’s entropy. When the system undergoes a phase transition from randomness to clustered behavior the curvature is smaller than that observed in the reverse phase transition. This asymmetric pattern relates to the emergence of hysteresis phenomenon, leading to an intrinsic arrow of time along the random field dynamics.

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“…Several random field models consider that the random variables can assume a finite and discrete number of states, such as the Ising [12] and the q-state Potts model [13]. In this paper, our focus is in the study of Gaussian random fields, where each variable can assume any value belonging the real line, that is, the set of possible states is infinite and continuous [14]. The main objective of this scientific investigation is to propose an information-geometric framework to understand and characterize the dynamics of Gaussian random fields defined on two-dimensional lattices.…”

Section: Introductionmentioning

confidence: 99%

On the curvatures of Gaussian random field manifolds

Levada

2021

Preprint

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Information geometry is concerned with the application of differential geometry concepts in the study of the parametric spaces of statistical models. When the random variables are independent and identically distributed, the underlying parametric space exhibit constant curvature, which makes the geometry hyperbolic (negative) or spherical (positive). In this paper, we derive closed-form expressions for the components of the first and second fundamental forms regarding pairwise isotropic Gaussian-Markov random field manifolds, allowing the computation of the Gaussian, mean and principal curvatures. Computational simulations using Markov Chain Monte Carlo dynamics indicate that a change in the sign of the Gaussian curvature is related to the emergence of phase transitions in the field. Moreover, the curvatures are highly asymmetrical for positive and negative displacements in the inverse temperature parameter, suggesting the existence of irreversible geometric properties in the parametric space along the dynamics (the curvature effect). Furthermore, these asymmetric changes in the curvature of the space induces an intrinsic notion of time in the evolution of the random field.

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Gaussian Random Fields

Cited by 4 publications

References 57 publications

On the Kullback-Leibler divergence between pairwise isotropic Gaussian-Markov random fields

On the Kullback-Leibler divergence between pairwise isotropic Gaussian-Markov random fields

The curvature effect in Gaussian random fields

On the curvatures of Gaussian random field manifolds

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