A lower bound on the average error of vector quantizers (Corresp.)

Conway, John H.; Sloane, N. J. A.

doi:10.1109/tit.1985.1056993

Cited by 70 publications

(16 citation statements)

References 22 publications

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“…Fig. 2 summarizes the best classical lattices [43-46, 33, 47] along with Conway and Sloane's conjectured lower bound [44]. With "classical" we mean lattices for which the normalized second moment has been reported previously.…”

Section: Methodsmentioning

confidence: 99%

“…For d = 8 , the only local minimum found is 11 Among the "best known" lattices, only the ones in dimensions 1-3 have been proven optimal. 12 The values were computed using a series expansion of the recursive integral equation in [44]. 13 The 5-dimensional locally optimal lattice can be obtained as the intersection of D 6 * and a hyperplane perpendicular to the vector 1 1 1 1 1 1 , , , , , ( ) T .…”

Section: A the Best Lattices Foundmentioning

confidence: 99%

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Optimization of lattices for quantization

Agrell

Eriksson

1998

IEEE Trans. Inform. Theory

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A training algorithm for the design of lattices for vector quantization is presented. The algorithm uses a steepest descent method to adjust a generator matrix, in the search for a lattice whose Voronoi regions have minimal normalized second moment. The numerical elements of the found generator matrices are interpreted and translated into exact values. Experiments show that the algorithm is stable, in the sense that several independent runs reach equivalent lattices. The obtained lattices reach as low second moments as the best previously reported lattices, or even lower. Specifically, we report lattices in 9 and 10 dimensions with normalized second moments of 0.0716 and 0.0708, respectively, and nonlattice tessellations in 7 and 9 dimensions with 0.0727 and 0.0711, which improves on previously known values. The new 9and 10-dimensional lattices suggest that Conway and Sloane's conjecture on the duality between the optimal lattices for packing and quantization might be false. A discussion of the application of lattices in vector quantizer design for various sources, uniform and nonuniform, is included.

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Section: Methodsmentioning

confidence: 99%

Section: A the Best Lattices Foundmentioning

confidence: 99%

Optimization of lattices for quantization

Agrell

Eriksson

1998

IEEE Trans. Inform. Theory

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“…We combine this result with the welfare loss for the uniform distribution in one dimension to obtain a lower bound on the welfare loss in higher dimensions. Conway and Sloane (1985), the space-…lling advantage is SF (d) e 6 . Clearly, S f U ; d = 1, and DP f U ; f U ; d = 1, i.e., there are no shape and dependence advantages for the i.i.d.…”

Section: Lower Bound On Welfare Lossmentioning

confidence: 99%

Nonlinear Pricing with Finite Information

Bergemann¹,

Shen

et al. 2015

SSRN Journal

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We analyze nonlinear pricing with …nite information. A seller o¤ers a menu to a continuum of buyers with a continuum of possible valuations. The menu is limited to o¤ering a …nite number of choices representing a …nite communication capacity between buyer and seller.We identify necessary conditions that the optimal …nite menu must satisfy, either for the socially e¢ cient or for the revenue-maximizing mechanism. These conditions require that information be bundled, or "quantized" optimally. We show that the loss resulting from using the n-item menu converges to zero at a rate proportional to 1=n 2 . We extend our model to a multi-product environment where each buyer has preferences over a d dimensional variety of goods. The seller is limited to o¤ering a …nite number n of d-dimensional choices. By using repeated scalar quantization, we show that the losses resulting from using the d-dimensional n-class menu converge to zero at a rate proportional to d=n 2=d . We introduce vector quantization and establish that the losses due to …nite menus are signi…cantly reduced by o¤ering optimally chosen bundles.

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“…[Zador 1982] developed an upper bound on C (d), or equivalently a lower bound on SF (d) using random quantization where the representation points are picked at random, and the partition is a Voronoi partition with respect to the set of representation points: [Conway and Sloane 1985] further showed that…”

Section: Advantages Of Vector Quantizationmentioning

confidence: 99%

Multi-dimensional mechanism design with limited information

Bergemann

Shen

et al. 2012

Proceedings of the 13th ACM Conference on Electronic Commerce

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We analyze a nonlinear pricing model with limited information. Each buyer can purchase a large variety, d, of goods. His preference for each good is represented by a scalar and his preference over d goods is represented by a d-dimensional vector. The type space of each buyer is given by a compact subset of R d + with a continuum of possible types. By contrast, the seller is limited to offer a finite number M of d-dimensional choices.We provide necessary conditions that the optimal finite menu of the social welfare maximizing problem has to satisfy. We establish an underlying connection to the theory of quantization and provide an estimate of the welfare loss resulting from the usage of the d-dimensional M -class menu. We show that the welfare loss converges to zero at a rate proportional to d/M 2/d . We show that in higher dimensions, a significant reduction in the welfare loss arises from an optimal partition of the d-dimensional type space that takes advantage of the correlation among the d parameters.

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A lower bound on the average error of vector quantizers (Corresp.)

Cited by 70 publications

References 22 publications

Optimization of lattices for quantization

Optimization of lattices for quantization

Nonlinear Pricing with Finite Information

Multi-dimensional mechanism design with limited information

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