Suchergebnisse

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. ; 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.

Proceeding of: 2017 IEEE International Symposium on Information Theory, Aachen, Germany, 25-30 June 2017 ; 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 →∞ to. This paper proposes an alternative definition of information dimension 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. It is demonstrated that both definitions are equivalent for stochastic processes that are ψ*-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. ; The work of Bernhard C. Geiger has 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 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 Spanish Ministerio de Economía y Competitividad 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.