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Universality of Logarithmic Loss in Fixed-Length Lossy Compression

Department of Electronic and Electrical Engineering, Hongik University, Seoul 04066, Korea
Entropy 2019, 21(6), 580; https://doi.org/10.3390/e21060580
Received: 8 April 2019 / Revised: 5 June 2019 / Accepted: 8 June 2019 / Published: 10 June 2019
(This article belongs to the Special Issue Bayesian Inference and Information Theory)
PDF [278 KB, uploaded 10 June 2019]

Abstract

We established a universality of logarithmic loss over a finite alphabet as a distortion criterion in fixed-length lossy compression. For any fixed-length lossy-compression problem under an arbitrary distortion criterion, we show that there is an equivalent lossy-compression problem under logarithmic loss. The equivalence is in the strong sense that we show that finding good schemes in corresponding lossy compression under logarithmic loss is essentially equivalent to finding good schemes in the original problem. This equivalence relation also provides an algebraic structure in the reconstruction alphabet, which allows us to use known techniques in the clustering literature. Furthermore, our result naturally suggests a new clustering algorithm in the categorical data-clustering problem.
Keywords: categorical data clustering; fixed-length lossy compression; logarithmic loss; rate-distortion categorical data clustering; fixed-length lossy compression; logarithmic loss; rate-distortion
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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No, A. Universality of Logarithmic Loss in Fixed-Length Lossy Compression . Entropy 2019, 21, 580.

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