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29 March 2024
 
  » arxiv » cs.IT/0411014

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A Theory of Lossy Compression for Individual Data
Nikolai K. Vereshchagin ; Paul M.B. Vitanyi ;
Date 7 Nov 2004
Subject Information Theory ACM-class: E.4, H.1.1 | cs.IT
AffiliationMoscow State Univ.), Paul M.B. Vitanyi (CWI and University of Amsterdam
AbstractWe develop rate-distortion theory for individual data with respect to general distortion measures, that is, a theory of lossy compression of individual data. This is applied to Euclidean distortion, Hamming distortion, Kolmogorov distortion, and Shannon-Fano distortion. We show that in all these cases for every function satisfying the obvious constraints there are data that have this function as their individual rate-distortion function. Shannon’s distortion-rate function over a random source is shown to be the pointswise asymptotic expectation of the individual distortion-rate functions we have defined. The great differences in the distortion-rate functions for individual non-random (that is, the aspects important to lossy compression) data we established were previously invisible and obliterated in the Shannon theory. The techniques are based on Kolmogorov complexity.
Source arXiv, cs.IT/0411014
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