Lossy compression of general random variables

Author:

Riegler Erwin1,Koliander Günther2,Bölcskei Helmut1

Affiliation:

1. ETH Zurich Chair for Mathematical Information Science, , Sternwartstrasse 7, 8092 Zurich, Switzerland

2. Austrian Academy of Sciences Acoustics Research Institute, , Wohllebengasse 12-14, 1040 Vienna, Austria

Abstract

AbstractThis paper is concerned with the lossy compression of general random variables, specifically with rate-distortion theory and quantization of random variables taking values in general measurable spaces such as, e.g. manifolds and fractal sets. Manifold structures are prevalent in data science, e.g. in compressed sensing, machine learning, image processing and handwritten digit recognition. Fractal sets find application in image compression and in the modeling of Ethernet traffic. Our main contributions are bounds on the rate-distortion function and the quantization error. These bounds are very general and essentially only require the existence of reference measures satisfying certain regularity conditions in terms of small ball probabilities. To illustrate the wide applicability of our results, we particularize them to random variables taking values in (i) manifolds, namely, hyperspheres and Grassmannians and (ii) self-similar sets characterized by iterated function systems satisfying the weak separation property.

Funder

WWTF

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis

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