S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions. SpringerVerlag, Berlin, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... mismatch, relative entropy, Kullback Leibler divergence 1 Introduction The optimal performance of high rate vector quantization using fixed rate codes was established in Zador s classic Bell Labs Technical Memo [35] and generalized and simplified by Bucklew and Wise [2] and Graf and Luschgy [14]. These results characterized the optimal rate distortion tradeo# of fixed dimension vector quantization as the rate or codebook size grows asymptotically large, in contrast to the Shannon rate distortion theory results characterizing the tradeo# for fixed rate when the dimension becomes ....

....and asymptotically large rate. Zador also developed the rate distortion tradeo#s for entropy constrained vector quantizers [35] but these results have only recently been generalized [20] to conditions of comparable generality to the fixed rate results of Bucklew and Wise [2] and Graf and Luschgy [14]. No rigorous variable rate mismatch results comparable to the fixed rate results of Bucklew [3] are known to the authors prior to those reported here. The primary goal of this paper is to establish a general variable rate mismatch result following the Lagrangian approach of [20] Applications to ....

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Springer, Lecture Notes in Mathematics, 1730, Berlin, 2000.

.... in adaptive vector quantizer design, where the R enyi entropy is more commonly called the Panter Dite factor and is related to the asymptotically optimal quantization cell density [11, 12] Furthermore, as empirical versions of vector quantization can be cast as geometric location problems [13], the asymptotics of adaptive VQ may be studied within the present framework of minimal Euclidean graphs. The outline of this report is as follows. In Section 2 we briefly review Redmond and Yukich s unifying framework of continuous quasi additive power weighted edge functionals. In Section 3 we ....

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg, 2000.

....may be found in [10] Most notably, Bucklew and Wise [2] demonstrated Zador s fixed rate result for rth power distortion measures of the form jjx Gamma yjj r , assuming only that E(jjXjj r ffi ) 1 for some ffi 0. Their result was subsequently simplified and elaborated by Graf and Luschgy [8]. Zador s entropyconstrained results, however, have not received similar attention in the literature. Zador formulated the entropy constrained problem as a minimization of average distortion over all quantizers with a constrained output entropy. Optimality properties and generalized Lloyd ....

....considered powers other than two. This is often stated loosely as ffi f (R) b(2; k)2 Zador s argument explicitly requires that his asymptotic result for fixed rate coding holds and that h(f) is finite. Zador s fixed rate conditions have been generalized through the years (see, e.g. 2] [8]) but his variable results have not been similarly extended. Furthermore, there are problems with Zador s proofs. In particular, as described in the proof of Lemma 2, Zador incorrectly assumes that a conditional entropy term is zero in his proof of his Corollary 3.3, an error which invalidates ....

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S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Springer, Lecture Notes in Mathematics, 1730, Berlin, 2000.

....(2.4) is exactly equivalent to that of finding coder and quantiser mappings such that q k 1 (X 0 , Z k 1 ) ##. 2. 6) The communication limited control problem is now in a form that bears a strong resemblance to standard mean mth power error (MmPE) asymptotic quantisation theory [7, 3, 8]. What distinguishes it are the facts that the quantisers q k , k W are recursive and that they and the multiplying product term are dependent on the Markov chain states. Nevertheless, quantisation arguments may still be applied to yield the following result: 3 Theorem 2.1. Suppose that the ....

S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions. Springer, 2000.

....may be found in [10] Most notably, Bucklew and Wise [2] demonstrated Zador s fixed rate result for rth power distortion measures of the form jjx Gamma yjj r , assuming only that E(jjXjj r ffi ) 1 for some ffi 0. Their result was subsequently simplified and elaborated by Graf and Luschgy [8]. Zador s entropyconstrained results, however, have not received similar attention in the literature. Zador formulated the entropy constrained problem as a minimization of average distortion over all quantizers with a constrained output entropy. Optimality properties and generalized Lloyd ....

....two. This is often stated loosely as ffi f (R) b(2; k)2 Gamma 2 k (R Gammah(f ) Zador s argument explicitly requires that his asymptotic result for fixed rate coding holds and that h(f) is finite. Zador s fixed rate conditions have been generalized through the years (see, e.g. 2] [8]) but his variable results have not been similarly extended. Furthermore, there are problems with Zador s proofs. In particular, as described in the proof of Lemma 2, Zador incorrectly assumes that a conditional entropy term is zero in his proof of his Corollary 3.3, an error which invalidates ....

[Article contains additional citation context not shown here]

Siegfried Graf and Harald Luschgy, Foundations of Quantization for Probability Distributions, Springer, Lecture Notes in Mathematics, 1730, Berlin, 2000.

....results may be found in [10] Most notably, Bucklew and Wise [2] demonstrated Zador s fixed rate result for rth power distortion measures of the form x y r , assuming only that E( X r # ) # for some # 0. Their result was subsequently simplified and elaborated by Graf and Luschgy [8]. Zador s entropyconstrained results, however, have not received similar attention in the literature. Zador formulated the entropy constrained problem as a minimization of average distortion over all quantizers with a constrained output entropy. Optimality properties and generalized Lloyd ....

....other than two. This is often stated loosely as # f (R) # b(2, k)2 2 k (R h(f) Zador s argument explicitly requires that his asymptotic result for fixed rate coding holds and that h(f) is finite. Zador s fixed rate conditions have been generalized through the years (see, e.g. 2] [8]) but his variable results have not been similarly extended. Furthermore, there are problems with Zador s proofs. In particular, as described in the proof of Lemma 2, Zador incorrectly assumes that a conditional entropy term is zero in his proof of his Corollary 3.3, an error which invalidates ....

[Article contains additional citation context not shown here]

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Springer, Lecture Notes in Mathematics, 1730, Berlin, 2000.

....are intervals, and the cells of a regular vector quantizer with a finite number of code points are convex polytopes. In this sense, the structure of optimal fixed rate quantizers for the squared error distortion measure (and to a certain extent for more general norm based distortion measures [5]) is relatively well understood. Moreover, for reasonable distortion measures and source distributions, the distortion of a quantizer satisfying the nearest neighbor condition is a continuous function of its code points, and so the existence of optimal fixed rate quantizers can be deduced using ....

....1=2 where the infimum is over all joint distributions of the pairs of random variables (X; Y ) such that X has distribution and Y has distribution . Then Delta is a metric on probability distributions with finite second moments which has been widely used in fixed rate quantization (see, e.g. [22, 6, 5]) For the squared error distortion measure, optimal fixed rate quantizer performance is a continuous function of the source distribution in this metric [6] that is, letting D f (N; denote the minimum mean squared distortion for a source with distribution of any quantizer with N code points, ....

S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions. Berlin, Heidelberg: Springer Verlag, 2000.

....optimal quantization problem of the probability measure . This question arises in various fields of applied mathematics (information theory, statistical clustering, stochastic algorithm theory, and has been extensively investigated during the past fifty years. Recently, Graf and Luschgy [11] have done a comprehensive book containing a rigorous mathematical treatment of the classical theory along with some investigations on new topics such as the optimal quantization for continuous singular probability measures. The survey of Gray and Neuhoff [12] 2 provides a detailed account of ....

....d ) n D ;n;r (y 1 ; yn ) D ;n;r Such a quantizer is called an optimal quantizer and D ;n;r is called the optimal distortion. Fact 2 : Asymptotics of the optimal distortion. If there exists 0 such that Z R d kuk r (du) 1. Then (Bucklew and Wise s theorem [6] see [11] for a correct proof) n r d D ;n;r Gamma J r;d kfk d d r where f is the density function of the absolutely continuous part of , where kfk d d r = GammaR f d= d r) Delta (d r) d and where J r;d is some constant depending only on r and d (see the Guersho s conjecture [13] for a ....

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Graf, S. and Luschgy, H. (1999) Foundations of Quantization for Probability Distributions. Forthcoming in Lecture notes in Mathematics, Springer.

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Graf S., Luschgy H., Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics n 1730, Springer, 2000, 230 p.

....: n : If (S 1 ; SN ) satis es the OSC then P (A A ) 0 if and are incomparable and moreover, P (A ) p (see [2] Lemma 3.3) 3. Statement of the main result For r 2 (0; 1) there exists a unique D r 2 (0; 1) satisfying N X i=1 (p i s r i ) Dr r Dr = 1 (see [3], Lemma 14.4) 3 3.1 Theorem Let (S 1 ; SN ) satisfy the OSC and let P be the self similar probability corresponding to (S 1 ; SN ; p) where p i 0 for i = 1; N . Let D r be as above. Then lim n 1 log n log e n;r = D r : 4. Proof of the main result In what follows q ....

.... n X k=1 (p (k) s r (k) t t r = 1: Since the sets A (1) A (n) are compact and pairwise disjoint we have = minfd(A (i) A (j) j1 i; j n; i 6= jg 0: There exisits a set k R d with card( k ) k and e k;r = Z d(x; k ) r dP (x) 1 r (see [3], Theorem 4.12) Set k = sup x2A d(x; k ) Then lim k 1 k = 0 (see [3] Lemma 13.8 and Lemma 6.1) Thus there exists an k 0 2 N with k 1 2 for all k k 0 . For k k 0 and i 2 f1; ng let k;i = fa 2 k jd(a; A (i) k g: 6 and n i (k) card( k;i ) Then n i (k) ....

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S. Graf { H. Luschgy, Foundations of Quantization for probability distributions, Lect. Notes Math. 1730, Springer 2000

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S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions. SpringerVerlag, Berlin, 2000.

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Graf S., Luschgy H. (2000): Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics n 1730, Springer, Berlin, 230 pp.

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S. Graf, H. Luschgy (2000), Foundations of quantization for probability distributions, Lecture Notes in Mathematics n 1730, Springer, Berlin, 230p.

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S. Graf, H. Luschgy (2000), Foundations of quantization for probability distributions, Lecture Notes in Mathematics n 1730, Springer, Berlin, 230p.

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Graf, S., Luschgy, H., Foundations of quantization of probability distributions, Lecture Notes Math. 1730, Springer, Berlin 2000

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S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions. Berlin, Heidelberg: Springer Verlag, 2000.

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S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions. Berlin, Heidelberg: Springer Verlag, 2000.

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S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics. Springer-Verlag, Berlin Heidelberg, 2000.

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S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Springer, Lecture Notes in Mathematics, 1730, Berlin, 2000.

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S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions. Springer Verlag, Berlin, Heidelberg, 2000.

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