Fastest Mixing Markov Chain on a Compact Manifold

Shiba Biswal, Karthik Elamvazhuthi, Spring Berman

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper, we address the problem of optimizing the convergence rate of a discrete-time Markov chain (DTMC), which evolves on a compact smooth connected manifold without boundary, to a specified target stationary distribution. This problem has been previously solved for a DTMC on a finite graph that converges to the uniform distribution. We consider arbitrary positive target measures that are supported on the entire state space of the system and are absolutely continuous with respect to the Riemannian volume. Similar to the previous work that addressed DTMCs on finite graphs, we pose the optimization problem in terms of maximizing the spectral gap of the operator that pushes forward measures, also known as the forward operator. Prior to formulating the optimization problem, we prove the existence of a forward operator that can stabilize the class of measures that we consider. In addition, we prove the existence of an optimal solution to our problem. The optimization problem admits an exact solution in the case where the manifold is a Lie group and the target measure is uniform. Lastly, we develop a numerical scheme for solving the optimization problem and validate our approach on a simulated system that evolves on a torus in{\mathbb{R}^3}

Original languageEnglish (US)
Title of host publication2019 IEEE 58th Conference on Decision and Control, CDC 2019
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages547-554
Number of pages8
ISBN (Electronic)9781728113982
DOIs
StatePublished - Dec 2019
Event58th IEEE Conference on Decision and Control, CDC 2019 - Nice, France
Duration: Dec 11 2019Dec 13 2019

Publication series

NameProceedings of the IEEE Conference on Decision and Control
Volume2019-December
ISSN (Print)0743-1546

Conference

Conference58th IEEE Conference on Decision and Control, CDC 2019
CountryFrance
CityNice
Period12/11/1912/13/19

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

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  • Cite this

    Biswal, S., Elamvazhuthi, K., & Berman, S. (2019). Fastest Mixing Markov Chain on a Compact Manifold. In 2019 IEEE 58th Conference on Decision and Control, CDC 2019 (pp. 547-554). [9029224] (Proceedings of the IEEE Conference on Decision and Control; Vol. 2019-December). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/CDC40024.2019.9029224