On the Information Dimension of Stochastic Processes

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dc.contributor.author Geiger, Bernhard C.
dc.contributor.author Koch, Tobias Mirco
dc.date.accessioned 2019-09-18T13:13:30Z
dc.date.available 2019-09-18T13:13:30Z
dc.date.issued 2019-10
dc.identifier.bibliographicCitation IEEE Transactions on Information Theory, 65(10), Pp. 6496 - 6518
dc.identifier.issn 0018-9448
dc.identifier.issn 1557-9654 (online)
dc.identifier.uri http://hdl.handle.net/10016/28825
dc.description.abstract In 1959, Rényi proposed the information dimension and the d-dimensional entropy to measure the information content of general random variables. This paper proposes a generalization of information dimension to stochastic processes by defining the information dimension rate as the entropy rate of the uniformly quantized stochastic process divided by minus the logarithm of the quantizer step size 1/m in the limit as m to infty. It is demonstrated that the information dimension rate coincides with the rate-distortion dimension, defined as twice the rate-distortion function R(D) of the stochastic process divided by -log (D) in the limit as D downarrow 0 . It is further shown that among all multivariate stationary processes with a given (matrix-valued) spectral distribution function (SDF), the Gaussian process has the largest information dimension rate and the information dimension rate of multivariate stationary Gaussian processes is given by the average rank of the derivative of the SDF. The presented results reveal that the fundamental limits of almost zero-distortion recovery via compressible signal pursuit and almost lossless analog compression are different in general.
dc.description.sponsorship The work of Bernhard C. Geiger has partly been funded by the Erwin Schrödinger Fellowship J 3765 of the Austrian Science Fund and by the German Ministry of Education and Research in the framework of an Alexander von Humboldt Professorship. The Know-Center is funded within the Austrian COMET Program - Competence Centers for Excellent Technologies - under the auspices of the Austrian Federal Ministry of Transport, Innovation and Technology, the Austrian Federal Ministry of Digital and Economic Affairs, and by the State of Styria. COMET is managed by the Austrian Research Promotion Agency FFG. The work of Tobias Koch has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement number 714161), from the 7th European Union Framework Programme under Grant 333680, from the Ministerio de EconomÍa y Competitividad of Spain under Grants TEC2013-41718-R, RYC-2014-16332, and TEC2016-78434-C3-3-R (AEI/FEDER, EU), and from the Comunidad de Madrid under Grant S2103/ICE-2845.
dc.format.extent 23
dc.language.iso eng
dc.publisher IEEE
dc.relation.ispartof http://hdl.handle.net/10016/26234
dc.relation.ispartof https://doi.org/10.3929/ethz-b-000245057
dc.rights © 2019 IEEE.
dc.subject.other Entropy
dc.subject.other Gaussian process
dc.subject.other Information dimension
dc.subject.other Rate-distortion dimension
dc.title On the Information Dimension of Stochastic Processes
dc.type article
dc.subject.eciencia Electrónica
dc.subject.eciencia Telecomunicaciones
dc.identifier.doi https://doi.org/10.1109/TIT.2019.2922186
dc.rights.accessRights openAccess
dc.relation.projectID info:eu-repo/grantAgreement/EC/H2020/714161/LOLITA
dc.relation.projectID info:eu-repo/grantAgreement/EC/FP7/333680/MobileNET
dc.relation.projectID Gobierno de España. TEC2013-41718-R
dc.relation.projectID Gobierno de España. RYC-2014-16332
dc.relation.projectID Gobierno de España. TEC2016-78434-C3-3-R
dc.relation.projectID Comunidad de Madrid. S2103/ICE-2845
dc.type.version acceptedVersion
dc.identifier.publicationfirstpage 6496
dc.identifier.publicationissue 10
dc.identifier.publicationlastpage 6518
dc.identifier.publicationtitle IEEE Transactions on Information Therory
dc.identifier.publicationvolume 65
dc.identifier.uxxi AR/0000023946
dc.contributor.funder European Commission
dc.contributor.funder Ministerio de Economía y Competitividad (España)
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