Shannon meets Lyapunov: Connections between information theory and dynamical systems

Tim Holliday, Peter Glynn, Andrea Goldsmith

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

1 Scopus citations

Abstract

This paper explores connections between Information Theory, Lyapunov exponents for products of random matrices, and hidden Markov models. Specifically, we will show that entropies associated with finite-state channels are equivalent to Lyapunov exponents. We use this result to show that the traditional prediction filter for hidden Markov models is not an irreducible Markov chain in our problem framework. Hence, we do not have access to many well-known properties of irreducible continuous state space Markov chains (e.g. a unique and continuous stationary distribution). However, by exploiting the connection between entropy and Lyapunov exponents and applying proof techniques from the theory of random matrix products we can solve a broad class of problems related to capacity and hidden Markov models. Our results provide strong regularity results for the non-irreducible prediction filter as well as some novel theoretical tools to address problems in these areas.

Original languageEnglish (US)
Title of host publicationProceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Pages1756-1763
Number of pages8
DOIs
StatePublished - 2005
Externally publishedYes
Event44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05 - Seville, Spain
Duration: Dec 12 2005Dec 15 2005

Publication series

NameProceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
Volume2005

Other

Other44th IEEE Conference on Decision and Control, and the European Control Conference, CDC-ECC '05
CountrySpain
CitySeville
Period12/12/0512/15/05

All Science Journal Classification (ASJC) codes

  • Engineering(all)

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