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The Dispersion of the Gauss-Markov Source

Tian, Peida and Kostina, Victoria (2018) The Dispersion of the Gauss-Markov Source. In: 2018 IEEE International Symposium on Information Theory (ISIT). IEEE , Piscataway, NJ, pp. 1490-1494. ISBN 978-1-5386-4780-6. http://resolver.caltech.edu/CaltechAUTHORS:20181126-141849980

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Abstract

The Gauss-Markov source produces U_i=aU_(i-1)+ Z_i for i ≥ 1, where U_0 = 0, |a| < 1 and Z_i ~ N(0, σ^2) are i.i.d. Gaussian random variables. We consider lossy compression of a block of n samples of the Gauss-Markov source under squared error distortion. We obtain the Gaussian approximation for the Gauss-Markov source with excess-distortion criterion for any distortion d > 0, and we show that the dispersion has a reverse waterfilling representation. This is the first finite blocklength result for lossy compression of sources with memory. We prove that the finite blocklength rate-distortion function R(n, d, ε) approaches the rate-distortion function R(d) as R(n, d, ε) = R(d)+√{[V(d)/n]}Q^(-1)(ε)+o([1/(√n)]), where V(d) is the dispersion, ε ∈ (0,1) is the excess-distortion probability, and Q^(-1) is the inverse of the Q-function. We give a reverse waterfilling integral representation for the dispersion V (d), which parallels that of the rate-distortion functions for Gaussian processes. Remarkably, for all 0 <; d ≤ σ2/(1+|a|)^2 ,R(n, d, c) of the Gauss-Markov source coincides with that of Zi, the i.i.d. Gaussian noise driving the process, up to the second-order term. Among novel technical tools developed in this paper is a sharp approximation of the eigenvalues of the covariance matrix of n samples of the Gauss-Markov source, and a construction of a typical set using the maximum likelihood estimate of the parameter a based on n observations.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/ISIT.2018.8437463DOIArticle
https://arxiv.org/abs/1804.09418arXivDiscussion Paper
http://resolver.caltech.edu/CaltechAUTHORS:20190610-093125985Related ItemJournal Article
ORCID:
AuthorORCID
Kostina, Victoria0000-0002-2406-7440
Additional Information:© 2018 IEEE. This research was supported in part by the National Science Foundation (NSF) under Grant CCF-1566567.
Funders:
Funding AgencyGrant Number
NSFCCF-1566567
Record Number:CaltechAUTHORS:20181126-141849980
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20181126-141849980
Official Citation:P. Tian and V. Kostina, "The Dispersion of the Gauss-Markov Source," 2018 IEEE International Symposium on Information Theory (ISIT), Vail, CO, 2018, pp. 1490-1494. doi: 10.1109/ISIT.2018.8437463
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:91185
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:26 Nov 2018 22:32
Last Modified:10 Jun 2019 17:02

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