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Entropy 2017, 19(6), 262; doi:10.3390/e19060262

Projection to Mixture Families and Rate-Distortion Bounds with Power Distortion Measures

Department of Computer Science and Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku-cho, Toyohashi 441-8580, Japan
Academic Editor: Geert Verdoolaege
Received: 4 May 2017 / Revised: 22 May 2017 / Accepted: 5 June 2017 / Published: 7 June 2017
(This article belongs to the Special Issue Information Geometry II)
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Abstract

The explicit form of the rate-distortion function has rarely been obtained, except for few cases where the Shannon lower bound coincides with the rate-distortion function for the entire range of the positive rate. From an information geometrical point of view, the evaluation of the rate-distortion function is achieved by a projection to the mixture family defined by the distortion measure. In this paper, we consider the β -th power distortion measure, and prove that β -generalized Gaussian distribution is the only source that can make the Shannon lower bound tight at the minimum distortion level at zero rate. We demonstrate that the tightness of the Shannon lower bound for β = 1 (Laplacian source) and β = 2 (Gaussian source) yields upper bounds to the rate-distortion function of power distortion measures with a different power. These bounds evaluate from above the projection of the source distribution to the mixture family of the generalized Gaussian models. Applying similar arguments to ϵ -insensitive distortion measures, we consider the tightness of the Shannon lower bound and derive an upper bound to the distortion-rate function which is accurate at low rates. View Full-Text
Keywords: rate-distortion function; Shannon lower bound; generalized Gaussian source; m-projection rate-distortion function; Shannon lower bound; generalized Gaussian source; m-projection
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Watanabe, K. Projection to Mixture Families and Rate-Distortion Bounds with Power Distortion Measures . Entropy 2017, 19, 262.

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