Theoretical investigations of an information geometric approach to complexity

Chaos: An Interdisciplinary Journal of Nonlinear Science

Mancini

2018

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Research on the use of information geometry (IG) in modern physics has witnessed significant advances recently. In this review article, we report on the utilization of IG methods to define measures of complexity in both classical and, whenever available, quantum physical settings. A paradigmatic example of a dramatic change in complexity is given by phase transitions (PTs). Hence, we review both global and local aspects of PTs described in terms of the scalar curvature of the parameter manifold and the components of the metric tensor, respectively. We also report on the behavior of geodesic paths on the parameter manifold used to gain insight into the dynamics of PTs. Going further, we survey measures of complexity arising in the geometric framework. In particular, we quantify complexity of networks in terms of the Riemannian volume of the parameter space of a statistical manifold associated with a given network. We are also concerned with complexity measures that account for the interactions of a given number of parts of a system that cannot be described in terms of a smaller number of parts of the system. Finally, we investigate complexity measures of entropic motion on curved statistical manifolds that arise from a probabilistic description of physical systems in the presence of limited information. The Kullback-Leibler divergence, the distance to an exponential family and volumes of curved parameter manifolds, are examples of essential IG notions exploited in our discussion of complexity. We conclude by discussing strengths, limits, and possible future applications of IG methods to the physics of complexity.

Section: B Complexitymentioning

confidence: 99%

Information geometric methods for complexity

Felice

Chaos: An Interdisciplinary Journal of Nonlinear Science

Mancini

2018

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“…For an extended review with more mathematical details and physical interpretations on the IGAC, we refer to the recent work appearing in Ref. [71].…”

Section: Discussionmentioning

confidence: 99%

An Information Geometric Perspective on the Complexity of Macroscopic Predictions Arising from Incomplete Information

Ali

Advances in Mathematical Physics

Gassner

et al. 2018

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Motivated by the presence of deep connections among dynamical equations, experimental data, physical systems, and statistical modeling, we report on a series of findings uncovered by the authors and collaborators during the last decade within the framework of the so-called Information Geometric Approach to Chaos (IGAC). The IGAC is a theoretical modeling scheme that combines methods of information geometry with inductive inference techniques to furnish probabilistic descriptions of complex systems in presence of limited information. In addition to relying on curvature and Jacobi field computations, a suitable indicator of complexity within the IGAC framework is given by the so-called information geometric entropy (IGE). The IGE is an information geometric measure of complexity of geodesic paths on curved statistical manifolds underlying the entropic dynamics of systems specified in terms of probability distributions. In this manuscript, we discuss several illustrative examples wherein our modeling scheme is employed to infer macroscopic predictions when only partial knowledge of the microscopic nature of a given system is available. Finally, we include comments on the strengths and weaknesses of the current version of our proposed theoretical scheme in our concluding remarks.

“…Substituting Eqs. (14) and (16) into Eq. (19), it happens that the Weyl anisotropy tensor components are identically zero.…”

Section: A the Fisher-rao Information Metricmentioning

confidence: 99%

“…For recent reviews on an information geometric perspective on the complexity of macroscopic predictions arising from incomplete information, we refer to Refs. [16][17][18].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the chaoticity (that is, temporal complexity) of such statistical models can be analyzed in terms of convenient indicators, such as: the signs of scalar and sectional curvatures of the statistical manifold, the asymptotic temporal behavior of Jacobi fields, the existence of Killing vectors, and the existence of a non-vanishing Weyl anisotropy tensor. In addition to these measures, complexity can also be quantified by means of the so-called information geometric entropy (IGE, [16][17][18]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Information geometric complexity of entropic motion on curved statistical manifolds under different metrizations of probability spaces

Gassner

Int. J. Geom. Methods Mod. Phys.

2019

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We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian geometric properties and entropic dynamical features of a Gaussian probability space where the two distinct dissimilarity measures between probability distributions are the Fisher-Rao information metric and the α-order entropy metric. In the former case, we observe an asymptotic linear temporal growth of the information geometric entropy (IGE) together with a fast convergence to the final state of the system. In the latter case, instead, we note an asymptotic logarithmic temporal growth of the IGE together with a slow convergence to the final state of the system. Finally, motivated by our findings, we provide some insights on a tradeoff between complexity and speed of convergence to the final state in our information geometric approach to problems of entropic inference.