- Rate distortion and denoising of individual data using Kolmogorov complexity
- IEEE Transactions on Information Theory
- Volume | Issue number
- 56 | 7
- Pages (from-to)
- Document type
- Interfacultary Research Institutes
- Institute for Logic, Language and Computation (ILLC)
We examine the structure of families of distortion balls from the perspective of Kolmogorov complexity. Special attention is paid to the canonical rate-distortion function of a source word which returns the minimal Kolmogorov complexity of all distortion balls containing that word subject to a bound on their cardinality. This canonical rate-distortion function is related to the more standard algorithmic rate-distortion function for the given distortion measure. Examples are given of list distortion, Hamming distortion, and Euclidean distortion. The algorithmic rate-distortion function can behave differently from Shannon's rate-distortion function. To this end, we show that the canonical rate-distortion function can and does assume a wide class of shapes (unlike Shannon's); we relate low algorithmic mutual information to low Kolmogorov complexity (and consequently suggest that certain aspects of the mutual information formulation of Shannon's rate-distortion function behave differently than would an analogous formulation using algorithmic mutual information); we explore the notion that low Kolmogorov complexity distortion balls containing a given word capture the interesting properties of that word (which is hard to formalize in Shannon's theory) and this suggests an approach to denoising.
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