DOCSLIB.ORG

Reparameterizing Distributions on Lie Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Reparameterizing Distributions on Lie Groups UvA-DARE (Digital Academic Repository) Reparameterizing Distributions on Lie Groups Falorsi, L.; de Haan, P.; Davidson, T.R.; Forré, P. Publication date 2019 Document Version Final published version Published in Proceedings of Machine Learning Research License Other Link to publication Citation for published version (APA): Falorsi, L., de Haan, P., Davidson, T. R., & Forré, P. (2019). Reparameterizing Distributions on Lie Groups. Proceedings of Machine Learning Research, 89, 3244-3253. https://arxiv.org/abs/1903.02958 General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:27 Sep 2021 Reparameterizing Distributions on Lie Groups Luca Falorsi1,2 Pim de Haan1,3 Tim R. Davidson1,2 Patrick Forré1 1University of Amsterdam 2Aiconic 3UC Berkeley Abstract non-trivial topology has had a longstanding tradition and giant impact on such fields as physics, mathemat- ics, and various engineering disciplines. One significant Reparameterizable densities are an important class describing numerous spaces of fundamental inter- way to learn probability distributions in a est is that of Lie groups, which are groups of symmetry deep learning setting. For many distributions transformations that are simultaneously differentiable it is possible to create low-variance gradient manifolds. Lie groups include rotations, translations, estimators by utilizing a ‘reparameterization scaling, and other geometric transformations, which trick’. Due to the absence of a general repa- play an important role in several application domains. rameterization trick, much research has re- Lie group elements are for example utilized to describe cently been devoted to extend the number the rigid body rotations and movements central in of reparameterizable distributional families. robotics, and form a key ingredient in the formula- Unfortunately, this research has primarily fo- tion of the Standard Model of particle physics. They cused on distributions defined in Euclidean also provide the building blocks underlying ideas in a space, ruling out the usage of one of the most plethora of mathematical branches such as holonomy in influential class of spaces with non-trivial Riemannian geometry, root systems in Combinatorics, topologies: Lie groups. In this work we define and the Langlands Program connecting geometry and a general framework to create reparameteri- number theory. zable densities on arbitrary Lie groups, and provide a detailed practitioners guide to fur- Many of the most notable recent results in machine ther the ease of usage. We demonstrate how learning can be attributed to researchers’ ability to com- to create complex and multimodal distribu- bine probability theoretical concepts with the power of tions on the well known oriented group of deep learning architectures, e.g. by devising optimiza- 3D rotations, SO(3), using normalizing flows. tion strategies able to directly optimize the parameters Our experiments on applying such distribu- of probability distributions from samples through back- tions in a Bayesian setting for pose estimation propagation. Perhaps the most successful instantiation on objects with discrete and continuous sym- of this combination, has come in the framework of Vari- metries, showcase their necessity in achieving ational Inference (VI) (Jordan et al., 1999), a Bayesian realistic uncertainty estimates. method used to approximate intractable probability densities through optimization. Crucial for VI is the ability to posit a flexible family of densities, and a way 1 INTRODUCTION to find the member closest to the true posterior by optimizing the parameters. Formulating observed data points as the outcomes of These variational parameters are typically optimized probabilistic processes, has proven to provide a useful using the evidence lower bound (ELBO), a lower bound framework to design successful machine learning mod- on the data likelihood. The two main approaches to ob- els. Thus far, the research community has drawn almost taining estimates of the gradients of the ELBO are the exclusively from results in probability theory limited to score function (Paisley et al., 2012; Mnih and Gregor, Euclidean and discrete space. Yet, expanding the set 2014) also known as REINFORCE (Williams, 1992), of possible spaces under consideration to those with a and the reparameterization trick (Price, 1958; Bonnet, 1964; Salimans et al., 2013; Kingma and Welling, 2013; Proceedings of the 22nd International Conference on Ar- Rezende et al., 2014). While various works have shown tificial Intelligence and Statistics (AISTATS) 2019, Naha, the latter to provide lower variance estimates, its use Okinawa, Japan. PMLR: Volume 89. Copyright 2019 by is limited by the absence of a general formulation for the author(s). Reparameterizing Distributions on Lie Groups all variational families. Although in recent years much as product. work has been done in extending this class of reparam- eterizable families (Ruiz et al., 2016; Naesseth et al., Lie Algebra, g: The Lie algebra g of a N dimen- 2017; Figurnov et al., 2018), none of these methods sional Lie group is its tangent space at the identity, explicitly investigate the case of distributions defined which is a vector space of N dimensions. We can see on non-trivial manifolds such as Lie groups. the algebra elements as infinitesimal generators, from which all other elements in the group can be created. The principal contribution of this paper is therefore For matrix Lie groups we can represent vectors v in to extend the reparameterization trick to Lie groups. the tangent space as matrices v . We achieve this by providing a general framework to × define reparameterizable densities on Lie groups, under Exponential Map, exp(·): The structure of the al- which the well-known Gaussian case of Kingma and gebra creates a map from an element of the algebra to Welling (2013) is recovered as a special instantiation. a vector field on the group manifold. This gives rise to This is done by pushing samples from the Lie algebra the exponential map, exp : g ! G which maps an alge- into the Lie group using the exponential map, and by bra element to the group element at unit length from observing that the corresponding density change can the identity along the flow of the vector field. The zero be analytically computed. We formally describe our vector is thus mapped to the identity. For compact con- approach using results from differential geometry and nected Lie groups, such as SO(3), the exponential map measure theory. is surjective. Often, the map is not injective, so the In the remainder of this work we first cover some pre- inverse, the log map, is multi-valued. The exponential liminary concepts on Lie groups and the reparameteri- map of matrix Lie groups is the matrix exponential. zation trick. We then proceed to present the general idea underlying our reparameterization trick for Lie Adjoint Representation, adx: The Lie algebra is groups (ReLie1), followed by a formal proof. Addition- equipped with with a bracket [·; ·]: g × g ! g, ally, we provide an implementation section2 where we which is bilinear. The bracket relates the structure study three important examples of Lie groups, deriving of the group to structure on the algebra. For exam- the reparameterization details for the n-Torus, TN , the ple, log(exp(x) exp(y)) can be expressed in terms of oriented group of 3D rotations, SO(3), and the group the bracket. The bracket of matrix Lie groups is the of 3D rotations and translations, SE(3). We conclude commutator of the algebra matrices. The adjoint rep- by creating complex and multimodal reparameterizable resentation of x 2 g is the matrix representation of the densities on SO(3) using a novel non-invertible normal- linear map adx : g ! g : y 7! [x; y]. izing flow, demonstrating applications of our work in both a supervised and unsupervised setting. 2.2 Reparameterization Trick The reparameterization trick (Price, 1958; Bonnet, 2 PRELIMINARIES 1964; Salimans et al., 2013; Kingma and Welling, 2013; Rezende et al., 2014) is a technique to simulate samples In this section we first cover a number of preliminary z ∼ q(z; θ) as z = T (; θ), where ∼ s() is indepen- concepts that will be used in the rest of this paper. dent from θ 4, and the transformation T (; θ) should be differentiable w.r.t. θ. It has been shown that this 2.1 Lie Groups and Lie Algebras generally results in lower variance estimates than score function variants, thus leading to more efficient and bet- Lie Group, G: A Lie group, G is a group that is also ter convergence results (Titsias and Lázaro-Gredilla, a smooth manifold. This means that we can, at least 2014; Fan et al., 2015). This reparameterization of in local regions, describe group elements continuously samples z, allows expectations w.r.t. q(z; θ) to be with parameters. The number of parameters equals rewritten as Eq(z;θ)[f(z)] = Es()[f(T (; θ))], thus mak- the dimension of the group. We can see (connected) ing it possible to directly optimize the parameters of a Lie groups as continuous symmetries where we can probability distribution through backpropagation. continuously traverse between group elements3.
Load more

Recommended publications
  • Quasi-Invariant Gaussian Measures for the Cubic Fourth Order Nonlinear Schrödinger Equation
    Probab. Theory Relat. Fields DOI 10.1007/s00440-016-0748-7 Quasi-invariant Gaussian measures for the cubic fourth order nonlinear Schrödinger equation Tadahiro Oh1,2 · Nikolay Tzvetkov3 Received: 4 August 2015 / Revised: 25 November 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com Abstract We consider the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we prove that the mean-zero Gaussian measures on Sobolev spaces s(T) > 3 H , s 4 , are quasi-invariant under the flow. Keywords Fourth order nonlinear Schrödinger equation · Biharmonic nonlinear Schrödinger equation · Gaussian measure · Quasi-invariance Mathematics Subject Classification 35Q55 Contents 1 Introduction ............................................... 1.1 Background ............................................. 1.2 Cubic fourth order nonlinear Schrödinger equation ........................ 1.3 Main result ............................................. 1.4 Organization of the paper ...................................... 2 Notations ................................................. 3 Reformulation of the cubic fourth order NLS .............................. B Tadahiro Oh [email protected] Nikolay Tzvetkov [email protected] 1 School of Mathematics, The University of Edinburgh, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK 2 The Maxwell Institute for the Mathematical Sciences, James Clerk Maxwell Building, The King’s Buildings, Peter Guthrie Tait Road,
    [Show full text]
  • A Brief on Characteristic Functions
    Missouri University of Science and Technology Scholars' Mine Graduate Student Research & Creative Works Student Research & Creative Works 01 Dec 2020 A Brief on Characteristic Functions Austin G. Vandegriffe Follow this and additional works at: https://scholarsmine.mst.edu/gradstudent_works Part of the Applied Mathematics Commons, and the Probability Commons Recommended Citation Vandegriffe, Austin G., "A Brief on Characteristic Functions" (2020). Graduate Student Research & Creative Works. 2. https://scholarsmine.mst.edu/gradstudent_works/2 This Presentation is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Graduate Student Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. A Brief on Characteristic Functions A Presentation for Harmonic Analysis Missouri S&T : Rolla, MO Presentation by Austin G. Vandegriffe 2020 Contents 1 Basic Properites of Characteristic Functions 1 2 Inversion Formula 5 3 Convergence & Continuity of Characteristic Functions 9 4 Convolution of Measures 13 Appendix A Topology 17 B Measure Theory 17 B.1 Basic Measure Theory . 17 B.2 Convergence in Measure & Its Consequences . 20 B.3 Derivatives of Measures . 22 C Analysis 25 i Notation 8 For all 8P For P-almost all, where P is a measure 9 There exists () If and only if U Disjoint union
    [Show full text]
  • ON the ISOMORPHISM PROBLEM for NON-MINIMAL TRANSFORMATIONS with DISCRETE SPECTRUM Nikolai Edeko (Communicated by Bryna Kra) 1. I
    DISCRETE AND CONTINUOUS doi:10.3934/dcds.2019262 DYNAMICAL SYSTEMS Volume 39, Number 10, October 2019 pp. 6001{6021 ON THE ISOMORPHISM PROBLEM FOR NON-MINIMAL TRANSFORMATIONS WITH DISCRETE SPECTRUM Nikolai Edeko Mathematisches Institut, Universit¨atT¨ubingen Auf der Morgenstelle 10 D-72076 T¨ubingen,Germany (Communicated by Bryna Kra) Abstract. The article addresses the isomorphism problem for non-minimal topological dynamical systems with discrete spectrum, giving a solution under appropriate topological constraints. Moreover, it is shown that trivial systems, group rotations and their products, up to factors, make up all systems with discrete spectrum. These results are then translated into corresponding results for non-ergodic measure-preserving systems with discrete spectrum. 1. Introduction. The isomorphism problem is one of the most important prob- lems in the theory of dynamical systems, first formulated by von Neumann in [17, pp. 592{593], his seminal work on the Koopman operator method and on dynamical systems with \pure point spectrum" (or \discrete spectrum"). Von Neumann, in particular, asked whether unitary equivalence of the associated Koopman opera- tors (\spectral isomorphy") implied the existence of a point isomorphism between two systems (\point isomorphy"). In [17, Satz IV.5], he showed that two ergodic measure-preserving dynamical systems with discrete spectrum on standard proba- bility spaces are point isomorphic if and only if the point spectra of their Koopman operators coincide, which in turn is equivalent to their
    [Show full text]
  • Sarnak's Möbius Disjointness for Dynamical Systems with Singular Spectrum and Dissection of Möbius Flow
    Sarnak’s Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow Houcein Abdalaoui, Mahesh Nerurkar To cite this version: Houcein Abdalaoui, Mahesh Nerurkar. Sarnak’s Möbius disjointness for dynamical systems with singular spectrum and dissection of Möbius flow. 2020. hal-02867657 HAL Id: hal-02867657 https://hal.archives-ouvertes.fr/hal-02867657 Preprint submitted on 15 Jun 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Sarnak’s M¨obius disjointness for dynamical systems with singular spectrum and dissection of M¨obius flow El Houcein El Abdalaoui Department of Mathematics Universite of Rouen, France and Mahesh Nerurkar Department of Mathematics Rutgers University, Camden NJ 08102 USA Abstract It is shown that Sarnak’s M¨obius orthogonality conjecture is fulfilled for the compact metric dynamical systems for which every invariant measure has singular spectra. This is accomplished by first establishing a special case of Chowla conjecture which gives a correlation between the M¨obius function and its square. Then a computation of W. Veech, followed by an argument using the notion of ‘affinity between measures’, (or the so called ‘Hellinger method’), completes the proof.
    [Show full text]
  • Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems Elise Janvresse, Emmanuel Roy, Thierry De La Rue
    Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems Elise Janvresse, Emmanuel Roy, Thierry de la Rue To cite this version: Elise Janvresse, Emmanuel Roy, Thierry de la Rue. Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems. Encyclopedia of Complexity and Systems Science, 2020, 978-3-642- 27737-5. 10.1007/978-3-642-27737-5_725-1. hal-02451104 HAL Id: hal-02451104 https://hal.archives-ouvertes.fr/hal-02451104 Submitted on 23 Jan 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. DYNAMICAL SYSTEMS OF PROBABILISTIC ORIGIN: GAUSSIAN AND POISSON SYSTEMS ELISE´ JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE Outline Glossary 1 1. Definition of the subject 3 2. Introduction 4 3. From probabilistic objects to dynamical systems 4 3.1. Gaussian systems 4 3.2. Poisson suspensions 5 4. Spectral theory 6 4.1. Basics of spectral theory 6 4.2. Fock space 7 4.3. Operators on a Fock space 7 4.4. Application to Gaussian and Poisson chaos 7 5. Basic ergodic properties 8 5.1. Ergodicity and mixing 8 5.2. Entropy, Bernoulli properties 9 6. Joinings, factors and centralizer 10 6.1.
    [Show full text]
  • An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions
    An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions Maria Gordina Abstract One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we cannot use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron- Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even though the main abstract result concerns quasi-invariance of a measure under a group action on a measure space. Keywords and phrases Quasi-invariance • Group action • Functional inequalities 1991 Mathematics Subject Classification. Primary 58G32 58J35; Secondary 22E65 22E30 22E45 58J65 60B15 60H05. 1 Introduction Our goal in this paper is to describe how a functional inequality can be used to prove quasi-invariance of certain measures in infinite dimensions. Even though the original argument was used in a geometric setting, we take a slightly different approach in this paper. Namely, we formulate a method that can be used to prove quasi- invariance of a measure under a group action. Such methods are useful in infinite dimensions when usually there is no natural reference measure such as the Lebesgue measure. At the same time quasi-invariance of measures is a useful tool in proving regularity results when it is reformulated as an integration by parts formula.
    [Show full text]
  • Incorporating Parameter Risk Into Derivatives Prices – Bid-Ask Pricing and Calibration
    Technische Universität München Lehrstuhl für Finanzmathematik Incorporating parameter risk into derivatives prices – bid-ask pricing and calibration Karl Friedrich Bannör Vollständiger Abdruck der von der Fakultät für Mathematik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzende: Univ.-Prof. Claudia Czado, Ph.D. Prüfer der Dissertation: 1. Univ.-Prof. Dr. Matthias Scherer 2. Univ.-Prof. Dr. Ralf Korn, Technische Universität Kaiserslautern 3. Lecturer Dr. Christoph Reisinger, University of Oxford/Vereinigtes Königreich Die Dissertation wurde am 16. Mai 2013 bei der Technischen Universität München eingereicht und durch die Fakultät für Mathematik am 26. Juli 2013 angenommen. To my beloved, all-time supporting wife Bianka, who made this work possible. Abstract In this thesis, a new methodology using convex risk measures is developed to incorporate parameter risk into prices of financial derivatives, provided that a distribution on the parameter space is given. In this context, weak continuity properties of convex risk measures w.r.t. the underlying probability measure on the parameter space are analyzed. Parameter risk arising from time series estimation is discussed in extensive numerical case studies and large-sample approximations for certain parameter risk-captured prices are stated. A technique to induce a parameter distribution in case of calibration to market prices is presented, allowing to conduct a comparison of parameter risk in different financial market models and of different exotic options. For the calibration to quoted bid-ask prices, a non-parametric calibration approach to broad classes of distortion risk measures is developed and a calibration to quoted bid-ask prices – comparing the non- parametric approach to given parametric suggestions – is assessed.
    [Show full text]
  • Semidefinite Approximations of Invariant Measures for Polynomial
    Semidefinite Approximations of Invariant Measures for Polynomial Systems Victor Magron1 Marcelo Forets1 Didier Henrion2;3 July 3, 2018 Abstract We consider the problem of approximating numerically the moments and the sup- ports of measures which are invariant with respect to the dynamics of continuous- and discrete-time polynomial systems, under semialgebraic set constraints. First, we address the problem of approximating the density and hence the support of an invariant measure which is absolutely continuous with respect to the Lebesgue mea- sure. Then, we focus on the approximation of the support of an invariant measure which is singular with respect to the Lebesgue measure. Each problem is handled through an appropriate reformulation into a linear op- timization problem over measures, solved in practice with two hierarchies of finite- dimensional semidefinite moment-sum-of-square relaxations, also called Lasserre hi- erarchies. Under specific assumptions, the first Lasserre hierarchy allows to approximate the moments of an absolutely continuous invariant measure as close as desired and to extract a sequence of polynomials converging weakly to the density of this measure. The second Lasserre hierarchy allows to approximate as close as desired in the Hausdorff metric the support of a singular invariant measure with the level sets of the Christoffel polynomials associated to the moment matrices of this measure. We also present some application examples together with numerical results for several dynamical systems admitting either absolutely
    [Show full text]
  • Arxiv:1811.08174V1 [Math.PR] 20 Nov 2018 Rcse Hc R Qiain Ne Oednmc.Tedifferen the Dynamics
    ERGODIC POISSON SPLITTINGS ELISE´ JANVRESSE, EMMANUEL ROY AND THIERRY DE LA RUE Abstract. In this paper we study splittings of a Poisson point process which are equivariant under a conservative transformation. We show that, if the Cartesian powers of this transformation are all ergodic, the only ergodic split- ting is the obvious one, that is, a collection of independent Poisson processes. We apply this result to the case of a marked Poisson process: under the same hypothesis, the marks are necessarily independent of the point process and i.i.d. Under additional assumptions on the transformation, a further applica- tion is derived, giving a full description of the structure of a random measure invariant under the action of the transformation. 1. Introduction Thinning and splitting are classical operations when studying point processes. Thinning consists in removing points according to some rule, whereas the related notion of splitting means decomposing the point process as the sum of several other point processes. It is well known that thinning a Poisson point process by choosing to remove points according to independent coin tosses yields a new Poisson process of lower (but proportional) intensity. Moreover, this procedure gives rise to a splitting of the original Poisson process into a sum of two independent Poisson processes. In recent years, some new results on thinnings of Poisson process have emerged. In particular, it is shown in [2] that it is possible to deterministically choose points from a homogeneous Poisson point process on R to get another homogeneous Pois- son process of lower intensity. Moreover, it is possible to proceed in a translation equivariant way.
    [Show full text]
  • Non-Autonomous Random Dynamical Systems: Stochastic Approximation and Rate-Induced Tipping
    Imperial College London Department of Mathematics Non-autonomous Random Dynamical Systems: Stochastic Approximation and Rate-Induced Tipping Michael Hartl Supervised by Prof Sebastian van Strien and Dr Martin Rasmussen A thesis presented for the degree of Doctor of Philosophy at Imperial College London. Declaration I certify that the research documented in this thesis is entirely my own. All ideas, theories and results that originate from the work of others are marked as such and fully referenced, and ideas originating from discussions with others are also acknowledged as such. 1 Copyright The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives license. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the license terms of this work. 2 Abstract In this thesis we extend the foundational theory behind and areas of application of non- autonomous random dynamical systems beyond the current state of the art. We generalize results from autonomous random dynamical systems theory to a non-autonomous realm. We use this framework to study stochastic approximations from a different point of view. In particular we apply it to study noise induced transitions between equilibrium points and prove a bifurcation result. Then we turn our attention to parameter shift systems with bounded additive noise. We extend the framework of rate induced tipping in deterministic parameter shifts for this case and introduce tipping probabilities.
    [Show full text]
  • Topics in Ergodic Theory: Notes 1
    Topics in ergodic theory: Notes 1 Tim Austin [email protected] http://www.math.brown.edu/ timaustin 1 Introduction In the first instance, a ‘dynamical system’ in mathematics is a mathematical object intended to capture the intuition of a ‘system’ in the real world that passes through some set of different possible ‘states’ as time passes. Although the origins of the subject lie in physical systems (such as the motion of the planets through different mutual positions, historically the first example to be considered), the mathematical study of these concepts encompasses a much broader variety of models arising in the applied sciences, ranging from the fluctuations in the values of different stocks in economics to the ‘dynamics’ of population sizes in ecology. The mathematical objects used to describe such a situation consist of some set X of possible ‘states’ and a function T : X −→ X specifying the (deterministic1) movement of the system from one state to the next. In order to make a useful model, one must endow the set X with at least a little additional structure and then assume that T respects that structure. Four popular choices, roughly in order of increasing specificity, are the following: • X a measurable space and T measurable — this is the setting of measurable dynamics; 1There are also many models in probability theory of ‘stochastic dynamics’, such as Markov processes for which T is replaced by a probability kernel, which is something akin to a ‘random- ized function’ – these lie outside the scope of this course. 1 • X a topological space and T continuous — topological dynamics; • X a smooth manifold and T a smooth map — smooth dynamics; • X a compact subset of C and T analytic — complex dynamics.
    [Show full text]
  • Hitting Time Statistics for Observations of Dynamical Systems
    HITTING TIME STATISTICS FOR OBSERVATIONS OF DYNAMICAL SYSTEMS JER´ OMEˆ ROUSSEAU Abstract. In this paper we study the distribution of hitting and return times for ob- servations of dynamical systems. We apply this results to get an exponential law for the distribution of hitting and return times for rapidly mixing random dynamical systems. In particular, it allows us to obtain an exponential law for random expanding maps, random circle maps expanding in average and randomly perturbed dynamical systems. 1. Introduction When trying to study the reality, experimentalists must often approximate their system or use a simplified model to get a lower dimensional system more suitable to analyze. In the same philosophy, when working on a high dimensional system, experimentalists generally are just interested in gauging different quantities (temperature, pressure, wind speed, wave height,...) or just know a measurement or observation of the system and tried to use theses observations to get informations on the whole system. Following these ideas, a Platonic formalism of dynamical system is given in [23]. More precisely, they want to determine what information on an attractor of a high dimensional dynamical system one can learn by knowing only its image under a function taking values in a lower dimensional space, called an observation. To investigate the statistical properties of dynamical systems, the quantitative description of Poincar´erecurrence behaviour plays an essential part and has been widely studied (see e.g. the review of Saussol [29]). Our aim in this article is to examine Poincar´erecurrence for observations of dynamical systems. The study of quantitative Poincar´erecurrence for observations began with the work of Boshernitzan [8] where it was shown that for a dynamical system (X,T,µ) and an observation f from X to a metric space (Y, d), if the α-dimensional Hausdorff measure is σ-finite on Y then lim inf n1/αd (f(x),f(T nx)) < ∞ for µ-almost every x.
    [Show full text]