The rate-distortion dimension of sets and measures

Kawabata, Tsutomu; Dembo, Amir

doi:10.1109/18.333868

Cited by 70 publications

(89 citation statements)

References 11 publications

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“…If the limits do not exist, we define the upper rate-distortion dimension dim R ({X t }) by replacing the limits with limits superior. Kawabata and Dembo showed that the rate-distortion dimension of a RV equals its information dimension [7]. This result can be generalized to stochastic processes.…”

Section: Information Dimension Rate Of Stochastic Processesmentioning

confidence: 82%

“…Rezagah et al showed in [5], [6] that d 0 (X) coincides, under certain conditions, with the rate-distortion dimension dim R ({X t }), defined as twice the rate-distortion function of the stochastic process {X t , t 2 Z} divided by log D in the limit as D # 0. This generalizes to stochastic processes the result by Kawabata and Dembo that the rate-distortion dimension of a random variable (RV) equals its information dimension [7].…”

Section: Introductionmentioning

confidence: 89%

“…For example, if the bandwidth of the PSD is 1/4, then we expect that half of the samples in {X t , t 2 Z} can be expressed as linear combinations of the other samples and, hence, do not contain information. In contrast, we show that the information dimension d By emulating the proof of [7,Lemma 3.2], we show that for any stochastic process {X t , t 2 Z}, the information dimension rate d({X t }) coincides with the rate-distortion dimension dim R ({X t }). This implies that d 0 (X) = dim R ({X t }) only for those stochastic processes for which d…”

Section: Introductionmentioning

confidence: 99%

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On the information dimension rate of stochastic processes

Geiger

Koch

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Abstract-Jalali and Poor ("Universal compressed sensing," arXiv:1406.7807v3, Jan. 2016) have recently proposed a generalization of Rényi's information dimension to stationary stochastic processes by defining the information dimension of the stochastic process as the information dimension of k samples divided by k in the limit as k ! 1. This paper proposes an alternative definition of information dimension as the entropy rate of the uniformlyquantized stochastic process divided by minus the logarithm of the quantizer step size 1/m in the limit as m ! 1. It is demonstrated that both definitions are equivalent for stochastic processes that are ⇤ -mixing, but that they may differ in general. In particular, it is shown that for Gaussian processes with essentially-bounded power spectral density (PSD), the proposed information dimension equals the Lebesgue measure of the PSD's support. This is in stark contrast to the information dimension proposed by Jalali and Poor, which is 1 if the process's PSD is positive on a set of positive Lebesgue measure, irrespective of its support size.

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Section: Information Dimension Rate Of Stochastic Processesmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

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On the information dimension rate of stochastic processes

Geiger

Koch

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…In fact [5] if X has a singular distribution, it is possible that d(X) < d(X). However, for the important class of self-similar singular distributions, the information dimension can be explicitly determined [9], [6]. For example, the Cantor distribution [10] has information dimension log 3 2.…”

Section: Rényi Information Dimensionmentioning

confidence: 99%

“…Rényi showed in [5] that under certain conditions for an absolutely continuous ndimensional random vector the information dimension is n. Hence he remarked that "... the geometrical (or topological) and information-theoretical concepts of dimension coincide for absolutely continuous probability distributions". However, the operational role of Rényi information dimension has not been addressed before except in the work of Kawabata and Dembo [6], which relates it to the rate-distortion function. In this paper we give a new operational characterization of Rényi information dimension as the fundamental limit of almost lossless data compression for analog sources under various regularity constraints of the encoder/decoder.…”

Section: Introductionmentioning

confidence: 99%

Fundamental limits of almost lossless analog compression

Verdú

2009

2009 IEEE International Symposium on Information Theory

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Abstract-In Shannon theory, lossless source coding deals with the optimal compression of discrete sources. Compressed sensing is a lossless coding strategy for analog sources by means of multiplication by real-valued matrices. In this paper we study almost lossless analog compression for analog memoryless sources in an information-theoretic framework, in which the compressor is not constrained to linear transformations but it satisfies various regularity conditions such as Lipschitz continuity. The fundamental limit is shown to be the information dimension proposed by Rényi in 1959.

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Double variational principle for mean dimension

Lindenstrauss

Tsukamoto

2019

Geom. Funct. Anal.

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We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

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The rate-distortion dimension of sets and measures

Cited by 70 publications

References 11 publications

On the information dimension rate of stochastic processes

On the information dimension rate of stochastic processes

Fundamental limits of almost lossless analog compression

Double variational principle for mean dimension

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