2,250 research outputs found
Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics
We study a probabilistic numerical method for the solution of both boundary
and initial value problems that returns a joint Gaussian process posterior over
the solution. Such methods have concrete value in the statistics on Riemannian
manifolds, where non-analytic ordinary differential equations are involved in
virtually all computations. The probabilistic formulation permits marginalising
the uncertainty of the numerical solution such that statistics are less
sensitive to inaccuracies. This leads to new Riemannian algorithms for mean
value computations and principal geodesic analysis. Marginalisation also means
results can be less precise than point estimates, enabling a noticeable
speed-up over the state of the art. Our approach is an argument for a wider
point that uncertainty caused by numerical calculations should be tracked
throughout the pipeline of machine learning algorithms.Comment: 11 page (9 page conference paper, plus supplements
Complexity Characterization in a Probabilistic Approach to Dynamical Systems Through Information Geometry and Inductive Inference
Information geometric techniques and inductive inference methods hold great
promise for solving computational problems of interest in classical and quantum
physics, especially with regard to complexity characterization of dynamical
systems in terms of their probabilistic description on curved statistical
manifolds. In this article, we investigate the possibility of describing the
macroscopic behavior of complex systems in terms of the underlying statistical
structure of their microscopic degrees of freedom by use of statistical
inductive inference and information geometry. We review the Maximum Relative
Entropy (MrE) formalism and the theoretical structure of the information
geometrodynamical approach to chaos (IGAC) on statistical manifolds. Special
focus is devoted to the description of the roles played by the sectional
curvature, the Jacobi field intensity and the information geometrodynamical
entropy (IGE). These quantities serve as powerful information geometric
complexity measures of information-constrained dynamics associated with
arbitrary chaotic and regular systems defined on the statistical manifold.
Finally, the application of such information geometric techniques to several
theoretical models are presented.Comment: 29 page
Dynamic isoperimetry and the geometry of Lagrangian coherent structures
The study of transport and mixing processes in dynamical systems is
particularly important for the analysis of mathematical models of physical
systems. We propose a novel, direct geometric method to identify subsets of
phase space that remain strongly coherent over a finite time duration. This new
method is based on a dynamic extension of classical (static) isoperimetric
problems; the latter are concerned with identifying submanifolds with the
smallest boundary size relative to their volume.
The present work introduces \emph{dynamic} isoperimetric problems; the study
of sets with small boundary size relative to volume \emph{as they are evolved
by a general dynamical system}. We formulate and prove dynamic versions of the
fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming
theorem and a dynamic Cheeger inequality. We introduce a new dynamic Laplacian
operator and describe a computational method to identify coherent sets based on
eigenfunctions of the dynamic Laplacian.
Our results include formal mathematical statements concerning geometric
properties of finite-time coherent sets, whose boundaries can be regarded as
Lagrangian coherent structures. The computational advantages of our new
approach are a well-separated spectrum for the dynamic Laplacian, and
flexibility in appropriate numerical approximation methods. Finally, we
demonstrate that the dynamic Laplacian operator can be realised as a
zero-diffusion limit of a newly advanced probabilistic transfer operator method
(Froyland, 2013) for finding coherent sets, which is based on small diffusion.
Thus, the present approach sits naturally alongside the probabilistic approach
(Froyland, 2013), and adds a formal geometric interpretation
Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC
We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm
for initializing pose graph optimization problems, arising in various scenarios
such as SFM (structure from motion) or SLAM (simultaneous localization and
mapping). TG-MCMC is first of its kind as it unites asymptotically global
non-convex optimization on the spherical manifold of quaternions with posterior
sampling, in order to provide both reliable initial poses and uncertainty
estimates that are informative about the quality of individual solutions. We
devise rigorous theoretical convergence guarantees for our method and
extensively evaluate it on synthetic and real benchmark datasets. Besides its
elegance in formulation and theory, we show that our method is robust to
missing data, noise and the estimated uncertainties capture intuitive
properties of the data.Comment: Published at NeurIPS 2018, 25 pages with supplement
First Order Feynman-Kac Formula
We study the parabolic integral kernel associated with the weighted Laplacian
and the Feynman-Kac kernels. For manifold with a pole we deduce formulas and
estimates for them and for their derivatives, given in terms of a Gaussian term
and the semi-classical bridge. Assumptions are on the Riemannian data.Comment: 31 pages, to appear in `Stochastic Processes and their Applications
Information geometric methods for complexity
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.Comment: review article, 60 pages, no figure
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