Firstly, we present the method from a regularization perspective. Using perturbations to target pivotal components in the models, we analyse datasets from political voting, finance and Twitter. … the book provides a nice introduction to a difficult subject that has many important applications.” (Marvin H. J. Gruber, Technometrics, Vol. Population-level scaling in ecological systems arises from individual growth and death with competitive constraints. Our payment security system encrypts your information during transmission. (gross), © 2020 Springer Nature Switzerland AG. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles. Quantification of uncertainty in predictions derived from such laws, and reduction of predictive uncertainty via data assimilation, remain an open challenge. Finally, numerical simulations utilizing neural network optimization demonstrate that tighter representations can result in significantly faster learning and more accurate estimation of divergences in both synthetic and real datasets (of more than 700 dimensions), often accelerated by nearly an order of magnitude. The book makes a wonderful reading for someone who enjoys the aesthetics of mathematics but do not expect to find in it anything that would help you with number crunching. Applications of the new matrix rearrangement inequality to the Schatten quasi-norms, the affine-invariant geometry and log-determinant divergences for positive definite matrices are also presented. We analyze how competing effects, such as specialization at later layers, may hide the positive transfer. The resulting algorithm is invariant to the parameterization of the belief distribution. to this metric is the distance. Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. The manifold possesses a Riemannian metric, two types of geodesics, and a divergence function. The latter has been formally described as a Euclidean space with an orthonormal basis featuring components that are suitable combinations of the original parts. Finally, we obtain the next iterate by following this direction according to the dual geometry induced by the Bregman potential. To this end, we model the time data by assuming two underlying probability distributions -- exponential and log-normal, and calculate some numerical characteristics for them. Transversal Levi-Civita connections for Riemannian foliations are generalized to the Lie groupoid/Lie algebroid case. These promotions will be applied to this item: Find all the books, read about the author and more. This work considers four dimensionality reduction methods: principal component analysis, sequential feature selection, ReliefF, and a novel feature ranking method. Fisher information and natural gradient provided deep insights and powerful tools to artificial neu-ral networks. This is highly reminiscent of the semiconvexity of entropy along interpolating lines of Wasserstein-2 transport in negative curvature spaces as established in [7], and we refer the reader to Remark 6 for more discussion. a voting configuration, stock and sector price movement, behaviour on social networks) determined by the set of parameters {J ij } indexed by ij, ... OT is not the only instance of a geometric framework for probability measures. Read : 170, Author : John Oprea Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. In this work we propose competitive mirror descent (CMD): a general method for solving such problems based on first order information that can be obtained by automatic differentiation. (gross), © 2020 Springer Nature Switzerland AG. The QoI is then optimized over this measure space. More generally, the sensitivity identifies pivotal components that precisely determine collective outcomes generated by a complex network of interactions. Please review prior to ordering, Introduces information geometry intuitively to readers without knowledge of differential geometry, Includes hot topics of applications to machine learning, signal processing, neural networks, and optimization, Applies information geometry to statistical inference and time-series analysis, ebooks can be used on all reading devices, Institutional customers should get in touch with their account manager, Usually ready to be dispatched within 3 to 5 business days, if in stock, The final prices may differ from the prices shown due to specifics of VAT rules. We propose to choose intermediate distributions using equal spacing in the moment parameters of our exponential family, which matches grid search performance and allows the schedule to adaptively update over the course of training. Our investigation is focused on deriving the relevant information metrics and their scalar curvatures on the space of equilibrium states for the corresponding gravitational backgrounds. We build on a minimal dynamical model of metabolic growth where the tension between individual growth and mortality determines population size distribution. These bang-bang processes comprise two steps: heating with the largest possible value of the driving and free cooling, i.e. Buy Information Geometry and Its Applications (Applied Mathematical Sciences) 1st ed. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. Machine Learning Shun-ichi Amari. 1 and the distributions arising from the Tsallis maxent principle [13]. Similar ideas can be used for the derivation of αand β-divergences [11]-, ... Information geometry provides a natural framework for measuring how sensitive collective properties are to change in component behaviour. In the terminology of information theory [9], P ({x}|t, β) expresses the likelihood of two parameters t and β for given data set of {x}. To get the free app, enter your mobile phone number. previous relation. Prime members enjoy Free Two-Day Shipping, Free Same-Day or One-Day Delivery to select areas, Prime Video, Prime Music, Prime Reading, and more. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. More precisely, we introduce and completely describe the compact closure of the moduli space of distributions of these statistics in several regimes. enable JavaScript in your browser. Here we develop the entropic dynamics of a system the state of which is described by a probability distribution. This method suggests to incorporate a recognition model as an auxiliary model for the efficient application of the natural gradient method in deep networks. Springer; 1st ed. We consider the parameter estimation problem of a probabilistic generative model prescribed using a natural exponential family of distributions. The Fisher information matrix (FIM) is fundamental for understanding the trainability of deep neural networks (DNN) since it describes the local metric of the parameter space. The method, which generalizes easily to many other cost functions, including the squared Euclidean distance, provides a novel combination of the Schrödinger problem approach due to C. Léonard and the related Brownian particle systems by Adams et al. Gaussian distributions are plentiful in applications dealing in uncertainty quantification and diffusivity. Exponential families have many appealing properties, such as the existence of conjugate priors and sufficient statistics, and a dually flat geometric structure. Both have their own merits for applications. Finally, we derive a doubly reparameterized gradient estimator which improves model learning and allows the TVO to benefit from more refined bounds. price for Spain I really like this book, and I would love to give it five stars. Read : 581, Author : Frank Nielsen He then states that the inner product w.r.t. Format : PDF, Mobi Our key idea is to treat each matrix as a probability distribution represented by a loglinear model on a partially ordered set (poset), which enables us to formulate rank reduction and balancing of a matrix as projection onto a statistical submanifold, which corresponds to the set of low-rank matrices or that of balanced matrices. Previously, DRO problems with a KL ambiguity set often result in tractable finite-dimensional reformulations [7,27,10]. These two geometries imply different natural gradients. Moreover, it will be made clear that this term also describes the underlying log-model volume, which we denote by log V, ... where (i) is the iteration index. Finally, we illustrate the resulting geometries with a numerical study. We prove that the Voronoi diagrams of the Fisher-Rao distance, the chi square divergence, and the Kullback-Leibler divergences all coincide with a hyperbolic Voronoi diagram on the corresponding Cauchy location-scale parameters, and that the dual Cauchy hyperbolic Delaunay complexes are Fisher orthogonal to the Cauchy hyperbolic Voronoi diagrams. As the uncertainty on the input distributions propagates to the QoI, an important consequence is that different choices of input distributions will lead to different values of the QoI.