Foundations of Quantization for Probability Distributions
Due to the rapidly increasing need for methods of data compression, quantization has become a flourishing field in signal and image processing and information theory. The same techniques are also used in statistics (cluster analysis), pattern recognition,
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1730
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Siegfried Graf Harald Luschgy
Foundations of Quantization for Probability Distributions
Springer
Authors Siegfried Graf Faculty for Mathematics and Computer Science University of Passau 94030 Passau, Germany E-mail: [email protected] Harald Luschgy FB IV, Mathematics University of Trier 54286 Trier, Germany E-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Graf, Siegfried: Foundations of quantization for probability distributions I Siegfried Graf ; Harald Luschgy. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer. 2000 (Lecture notes in mathematics; 1730) ISBN 3-540-67394-6
Mathematics Subject Classification (2000): 60Exx, 62H30, 28A80, 90B05, 94A29 ISSN 0075-8434 ISBN 3-540-67394-6 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law, Springer- Verlag is a company in the BertelsmannSpringer publishing group. © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author Printed on acid-free paper SPIN: 10724973 41/3143/du
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Contents List of Figures
VIII
List of Tables
IX
Introduction
1
I
General properties of the quantization for probability distributions
7
1
Voronoi partitions. . .
7
1.1
General norms.
7
1.2
Euclidean norms
16
2
Centers and moments of probability distributions
20
2.1
Uniqueness and characterization of centers
20
2.2
Moments of balls .
26
3
The quantization problem
30
4
Basic properties of optimal quantizers .
37
4.1
Stationarity and existence
37
4.2
The functional Vn,T . . . .
48
4.3
Quantization error for ball packings .
50
4.4
Examples...............
52
4.5
Stability properties and empirical versions
57
5
Uniqueness and optimality in one dimension
64
5.1
Uniqueness.....
64
5.2
Optimal Quantizers .
66
II Asymptotic quantization for nonsingular probability distributions 6
Asymptotics for the quantization error . . . . . . . . . . . . . . . ..
77 77
Contents
vi 7
8
9
Asymptotically optimal quantizers .
93
7.1
Mi
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