A result on multidimensional quantization

Hall, Eric B.; Wise, Gary L.

doi:10.1090/S0002-9939-1993-1129878-X

A result on multidimensional quantization
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by Eric B. Hall and Gary L. Wise PDF

Proc. Amer. Math. Soc. 118 (1993), 609-613 Request permission

Abstract:

For any integer $N > 1$, a probability space, a Gaussian random vector $X$ defined on the space with a positive definite covariance matrix, and an $N$-level quantizer $Q$ are presented such that the random vector $Q(X)$ takes on each of the $N$ values in its range with equal probability and such that $X$ and $Q(X)$ are independent.

References

Allen Gersho, Asymptotically optimal block quantization, IEEE Trans. Inform. Theory 25 (1979), no. 4, 373–380. MR 536229, DOI 10.1109/TIT.1979.1056067
Robert M. Gray and E. D. Karnin, Multiple local optima in vector quantizers, IEEE Trans. Inform. Theory 28 (1982), no. 2, 256–261. MR 651822, DOI 10.1109/TIT.1982.1056471
R. M. Gray, J. C. Kieffer, and Y. Linde, Locally optimal block quantizer design, Inform. and Control 45 (1980), no. 2, 178–198. MR 584832, DOI 10.1016/S0019-9958(80)90313-7
Paul R. Halmos, Naive set theory, Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1974. Reprint of the 1960 edition. MR 0453532

An algorithm for vector quantizer design

COM-28

David Pollard, Quantization and the method of $k$-means, IEEE Trans. Inform. Theory 28 (1982), no. 2, 199–205. MR 651814, DOI 10.1109/TIT.1982.1056481
Michael J. Sabin and Robert M. Gray, Global convergence and empirical consistency of the generalized Lloyd algorithm, IEEE Trans. Inform. Theory 32 (1986), no. 2, 148–155. MR 838406, DOI 10.1109/TIT.1986.1057168
Max Shiffman, Measure-theoretic properties of nonmeasurable sets, Pacific J. Math. 138 (1989), no. 2, 357–389. MR 996206

A vector quantizer for the Laplacian source

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