Methods Of Information Geometry Research Articles

From CFTs to theories with Bondi-Metzner-Sachs symmetries: Complexity and out-of-time-ordered correlators

We probe the contraction from $2d$ relativistic CFTs to theories with Bondi-Metzner-Sachs (BMS) symmetries, or equivalently Conformal Carroll symmetries, using diagnostics of quantum chaos. Starting from an Ultrarelativistic limit on a relativistic scalar field theory and following through at the quantum level using an oscillator representation of states, one can show the CFT$_2$ vacuum evolves smoothly into a BMS$_3$ vacuum in the form of a squeezed state. Computing circuit complexity of this transmutation using the covariance matrix approach shows clear divergences when the BMS point is hit or equivalently when the target state becomes a boundary state. We also find similar behaviour of the circuit complexity calculated from methods of information geometry. Furthermore, we discuss the hamiltonian evolution of the system and investigate Out-of-time-ordered correlators (OTOCs) and operator growth complexity, both of which turn out to scale polynomially with time at the BMS point.

Unsupervised manifold learning with polynomial mapping on symmetric positive definite matrices

In this paper, an unsupervised manifold learning algorithm with polynomial mapping on the symmetric positive-definite (SPD) matrix manifold is introduced by matrix information geometry method for data dimensional reduction. Firstly, the mathematical knowledge about the SPD matrix manifold is presented including the metric, geodesic and submanifold. And then, the high dimensional information coordinates are given by different SPD matrix data for constructing the polynomial kernel matrix, weight matrix and sparsity preserving matrix. Next, the manifold learning algorithm on the SPD matrix manifold is proposed by polynomial mapping with geodesic distance. Finally, comparing with some conventional methods in terms of accuracy rate and time cost, the preliminary analysis results indicate that the proposed approach is able to offer a consistent and comprehensive method to realize the SPD matrix data dimensional reduction.

Ship Target Detection in SAR Imagery Based on Maximum Eigenvalue Detector

Ship target detection in synthetic aperture radar (SAR) imagery is of great significance in the field of ocean monitoring. Classical constant false alarm rate (CFAR) detectors and emerging information geometry methods are model-driven essentially, requiring precise modeling of the sea clutter distribution. In the complex and changeable ocean scenes, the performance of these two types of detectors is limited. To solve this problem, a ship target detection algorithm in SAR imagery based on the maximum eigenvalue of the sample covariance matrix is proposed in this letter. Without seeking the distribution model of clutter backgrounds, the difference between the target and the clutter background is fully captured by constructing the sample covariance matrix, and its maximum eigenvalue is utilized as the test statistic. Experimental results on measured SAR images show that the proposed method achieves better detection performance and faster calculation speed compared with the existing typical methods.

IGAGCN: Information geometry and attention-based spatiotemporal graph convolutional networks for traffic flow prediction

In this study, a novel spatiotemporal graph convolutional networks model is proposed for traffic flow prediction in urban road networks by fully considering an information geometry approach and attention-based mechanism. Accurate traffic flow prediction in real urban road networks is challenging due to the presence of dynamic spatiotemporal data and external factors in the urban environment. Moreover, the dynamic spatial and temporal dependencies of urban traffic flow data are very important for predicting traffic flow, and it has been shown that a recent attention mechanism has a relatively good ability to capture these dynamic dependencies, which are not fully considered by most existing algorithms. Therefore, in the novel model abbreviated as IGAGCN, the information geometry method is utilized to determine the dynamic data distribution difference between different sensors. The attention mechanism is employed with the information geometry method, in which a matrix is derived by analyzing the distributions of sensor data, and the spatiotemporal dynamic connections in traffic flow data features are better at capturing the spatial dependencies of traffic between different sensors in urban road networks. Furthermore, a parallel sub-model architecture is proposed to consider long time spans, where each dilated causal convolution sub-model is applied to short time spans. Two well-known data sets were employed to demonstrate that our proposed method obtains better performance and is better at capturing the dynamic spatial dependencies of traffic than the existing only-attention-based models. In addition a real-world urban road network in Shenzhen, China, was studied to test and verify the proposed model.

Information Geometry of the Exponential Family of Distributions with Progressive Type-II Censoring.

In geometry and topology, a family of probability distributions can be analyzed as the points on a manifold, known as statistical manifold, with intrinsic coordinates corresponding to the parameters of the distribution. Consider the exponential family of distributions with progressive Type-II censoring as the manifold of a statistical model, we use the information geometry methods to investigate the geometric quantities such as the tangent space, the Fisher metric tensors, the affine connection and the -connection of the manifold. As an application of the geometric quantities, the asymptotic expansions of the posterior density function and the posterior Bayesian predictive density function of the manifold are discussed. The results show that the asymptotic expansions are related to the coefficients of the -connections and metric tensors, and the predictive density function is the estimated density function in an asymptotic sense. The main results are illustrated by considering the Rayleigh distribution.

The geometric structure on a degradation model with application to optimal design under a cost constraint

Information geometry has been attracting considerable attention in different scientific fields including information theory, neural networks, machine learning, and statistical physics. In reliability and survival analysis, methods of information geometry are employed to discuss the geometry on a reliability model. Most of the existing work of information geometry in reliability analysis focused on time-to-failure data. For highly reliable devices, it is difficult to obtain failure data in a reasonable period of time, and hence degradation measurements are used to extrapolate the failure time. In this paper, we investigate the geometry on a statistical manifold induced by the Wiener degradation process with nonlinear drift and diffusion coefficients, where the drift parameter is assumed to be a random variable to incorporate the unit-to-unit characteristics. The Fisher information metric, Amari–Chentsov tensor and α-connection on the manifold for the degradation model are discussed. As an application in the design of engineering experiments, the information metric is employed to develop the optimal design of degradation experiments under a cost constraint. Monte Carlo simulation and a numerical study are used to illustrate the methodologies developed in this paper.

Application of joint domain localised matrix CFAR detector for HFSWR

Target detection is one of the most important parts of high-frequency surface wave radar (HFSWR) signal processing, to find targets in noise or clutter and obtain targets’ information. However, some factors, such as the influence of clutter and the increase of detection range, will degrade the signal-to-clutter ratio (SCR). In the low SCR scenario, the classical detector of HFSWR, which uses the amplitude of receipt signal only, could hardly detect targets. To solve this problem, this study proposes a novel detector called joint domain localised matrix constant false-alarm rate (CFAR) detector based on receipt signal's multi-dimensional information. This detector employs joint domain localised algorithm to get signal information in angle and Doppler domain, and use information geometry method to map them to Hermitian positive-definite (HPD) matrix space which can be depicted as Riemannian manifold. Then, based on HPD matrix, in the range domain the matrix CFAR detector is built to detect targets. The experiments’ results verify that the detector can improve radar detection performance effectively.

Geometry on the statistical manifold induced by the degradation model with soft failure data

Information geometry discusses the properties of a statistical manifold and is useful for different fields involving neural networks, signal processing, machine learning, optimization and statistics. The methods of information geometry are also employed to discuss the geometry on a statistical manifold induced by the reliability model and lifetime testing, where the time-to-failure data are available for analyzing. For highly reliable products, however, it is difficult to obtain failure data in a reasonable period of time. For some products there is a gradual loss of performance, then it is possible to derive degradation measurements over times. This paper investigates the geometry on a statistical manifold induced by the degradation model with few or no failure data. The statistical manifold is constructed based on the degradation model. The Fisher information metric, Amari–Chentsov structure, affine connection and α -connection on the manifold are discussed. Taking the linear model as an example, the main results are illustrated, where we find that the geometry quantities are closely related to the degradation threshold value, parameters of the model and the Euler’s constant. The parameters are estimated by using the multi-step estimation method, and numerical results are reported. • The probability density function of the soft failure data is obtained in a closed form. • The statistical manifold is constructed based on the degradation model. • The geometry on the statistical manifold is discussed. • The parameters are estimated by using multi-step estimation methods.

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

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.

Neyman–Pearson Test for Zero-Rate Multiterminal Hypothesis Testing

The problem of zero-rate multiterminal hypothesis testing is revisited from the perspective of information-spectrum approach and finite blocklength analysis. A Neyman–Pearson-like test is proposed and its non-asymptotic performance is clarified, for a short block length, it is numerically determined that the proposed test is superior to the previously reported Hoeffding-like test proposed by Han–Kobayashi. For a large deviation regime, it is shown that our proposed test achieves an optimal trade-off between the type I and type II exponents presented by Han–Kobayashi. Among the class of symmetric (type-based) testing schemes, when the type I error probability is non-vanishing, the proposed test is optimal up to the second-order term of the type II error exponent; the latter term is characterized in terms of the variance of the projected relative entropy density. The information geometry method plays an important role in the analysis as well as the construction of the test.

Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes.

The relevance of the concept of Fisher information is increasing in both statistical physics and quantum computing. From a statistical mechanical standpoint, the application of Fisher information in the kinetic theory of gases is characterized by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case. From a quantum mechanical standpoint, the output state in Grover's quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. Additionally, Grover's algorithm is specified by constant Fisher information. In this paper, we present an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and power-law decay. Furthermore, for each case, we compute both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts. Finally, we briefly comment on the possibility of using the proposed methods of information geometry to help identify a suitable trade-off between speed and thermodynamic efficiency in quantum search algorithms.

Information Geometry of Generalized Bayesian Prediction Using $\alpha$ -Divergences as Loss Functions

In this paper, the methods of information geometry are employed to investigate a generalized Bayes rule for prediction. Taking ${\alpha }$ -divergences as the loss functions, optimality, and asymptotic properties of the generalized Bayesian predictive densities are considered. We show that the Bayesian predictive densities minimize a generalized Bayes risk. We also find that the asymptotic expansions of the densities are related to the coefficients of the ${\alpha }$ -connections of a statistical manifold. In addition, we discuss the difference between two risk functions of the generalized Bayesian predictions based on different priors. Finally, using the non-informative priors (i.e., Jeffreys and reference priors), uniform prior, and conjugate prior, two examples are presented to illustrate the main results.

A novel information geometry method for estimating parameters of the Weibull wind speed distribution

A novel information geometry method for estimating parameters of the Weibull wind speed distribution

Methods of Information Geometry to model complex shapes

In this paper, a new statistical method to model patterns emerging in complex systems is proposed. A framework for shape analysis of 2− dimensional landmark data is introduced, in which each landmark is represented by a bivariate Gaussian distribution. From Information Geometry we know that Fisher-Rao metric endows the statistical manifold of parameters of a family of probability distributions with a Riemannian metric. Thus this approach allows to reconstruct the intermediate steps in the evolution between observed shapes by computing the geodesic, with respect to the Fisher-Rao metric, between the corresponding distributions. Furthermore, the geodesic path can be used for shape predictions. As application, we study the evolution of the rat skull shape. A future application in Ophthalmology is introduced.

信息几何理论与应用研究进展

Information geometry is the fundamental and cutting-edge discipline that explores statistical problems on Riemannian manifolds of probability distributions using the methods of differential geometry. It has been identified as the second generation of modern information theory pioneered by Shannon, and exhibits great potential in developing the field of information science and systems theory. This paper begins defining the scientific content of information geometry from the intrinsic geometrical structures of parameterized families of probability distributions as well as geometric properties of information; it points out theoretical advantages and methodological innovations of information geometry compared with classical statistics and information theory. Next, the paper briefly introduces connections between information geometry and differential geometry, and elaborates the history of information geometry as well as its applications to various areas such as neural networks, statistical inference, communications and coding, systems and control theory, physics and medical imaging, among others. In particular, the applications of information geometry to signal processing, including the latest results of geometric methods of signal detection, nonlinear parameter estimation, and filtering, are comprehensively introduced. The basic ideas and methods of information geometry are also summarized. Finally, by viewing prospects for the development of information geometry, several open problems of information geometry in applications to signal processing are proposed.

On the Information Geometry of Audio Streams With Applications to Similarity Computing

This paper proposes methods for information processing of audio streams using methods of information geometry. We lay the theoretical groundwork for a framework allowing the treatment of signal information as information entities, suitable for similarity and symbolic computing on audio signals. The theoretical basis of this paper is based on the information geometry of statistical structures representing audio spectrum features, and specifically through the bijection between the generic families of Bregman divergences and that of exponential distributions. The proposed framework, called Music Information Geometry, allows online segmentation of audio streams to metric balls where each ball represents a quasi-stationary continuous chunk of audio, and discusses methods to qualify and quantify information between entities for similarity computing. We define an information geometry that approximates a similarity metric space, redefine general notions in music information retrieval such as similarity between entities, and address methods for dealing with nonstationarity of audio signals. We demonstrate the framework on two sample applications for online audio structure discovery and audio matching.

Methods of Information Geometry (Amari, S. and Nagaoka, H.; 2000) [Book review

This book presents the state of the art, the current best practices and the innovative techniques of the methods of information geometry. The first half of the book presents a comprehensive introduction to the mathematical foundation of information geometry, including preliminaries from differential geometry, the geometry of manifolds or probability distributions, and the general theory of dual affine connections. The second half describes many applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. This is a well-written book, original and highly relevant from which to learn methods and results.

Linear Gauge and the Linearization Problem for Webs

Regarding the theory of foliation, some aspects of the theory of Riemannian foliations have been brought in completion by the Molino theory. Such a structure is defined by some finite dimensional Lie subalgebra of the Lie algebra of transverse vector fields. The problem I am interested in is more modeste. It is to get sufficient conditions for a smooth manifold admitting foliations with transverse (pseudo) Riemannian metrics. The investigation is inspired by both methods of information geometry and the Hessian geometry.

The geometry of prior selection

This contribution is devoted to the selection of prior in a Bayesian learning framework. There is an extensive literature on the construction of non-informative priors and the subject seems far from a definite solution [Kass and Wasserman, Formal rules for selecting prior distributions: a review and annotated bibliography, Technical Report No. 583, Department of Statistics, Carnegie Mellon University, 1994]. We consider this problem in the light of the recent development of information geometric tools [Amari and Nagaoka, Methods of information geometry, in: Translations of Mathematical Monographs, AMS, vol. 191, Oxford University Press, Oxford, 2000]. The differential geometric analysis allows the formulation of the prior selection problem in a general manifold valued set of probability distributions. In order to construct the prior distribution, we propose a criteria expressing the trade off between decision error and uniformity constraint. The solution has an explicit expression obtained by variational calculus. In addition, it has two important invariance properties: invariance to the dominant measure of the data space and also invariance to the parametrization of a restricted parametric manifold. We show how the construction of a prior by projection is the best way to take into account the restriction to a particular family of parametric models. For instance, we apply this procedure to autoparallel restricted families. Two practical examples illustrate the proposed construction of prior. The first example deals with the learning of a mixture of multivariate Gaussians in a classification perspective. We show in this learning problem how the penalization of likelihood by the proposed prior eliminates the degeneracy occurring when approaching singularity points. The second example treats the blind source separation problem.