Intrinsic Stationarity for Vector Quantization: Foundation of Dual Quantization

Pagès, Gilles; Wilbertz, Benedikt

doi:10.1137/110827041

Cited by 17 publications

(66 citation statements)

References 18 publications

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“…since it holds for for any grid (see [64]). These provides a two-sided bound can be numerically implemented.…”

Section: Application To Convex Functionsmentioning

confidence: 98%

“…[25] and the references therein). More recently, a new concept of quantization (dual quantization, see [64]) has refined this point of view by switching from Voronoi diagrams to a direct approach based on optimized Delaunay triangulations. The resulting grids are better adapted to deterministic numerical analysis methods in medium dimensions.…”

Section: Toward Automatic Meshingmentioning

confidence: 99%

“…Unfortunately, it is satisfied only by few quantizers. A new notion of quantization, called dual quantization has been recently developed (see [64] for a theoretical introduction and [63,65] for applications to Numerical Probability) in which a reverse stationarity property is satisfied by all dual quantizers. Typically for dual quantization, one has…”

Section: Remarksmentioning

confidence: 99%

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Introduction to vector quantization and its applications for numerics

Pagès

2015

ESAIM: Proc.

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Abstract. We present an introductory survey to optimal vector quantization and its first applications to Numerical Probability and, to a lesser extent to Information Theory and Data Mining. Both theoretical results on the quantization rate of a random vector taking values in R d (equipped with the canonical Euclidean norm) and the learning procedures that allow to design optimal quantizers (CLV Q and Lloyd's procedures) are presented. We also introduce and investigate the more recent notion of greedy quantization which may be seen as a sequential optimal quantization. A rate optimal result is established. A brief comparison with Quasi-Monte Carlo method is also carried out.

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“…since it holds for for any grid (see [64]). These provides a two-sided bound can be numerically implemented.…”

Section: Application To Convex Functionsmentioning

confidence: 98%

Section: Toward Automatic Meshingmentioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

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Introduction to vector quantization and its applications for numerics

Pagès

2015

ESAIM: Proc.

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“…In addition, alternative quantization approaches, such as the so-called dual quantization and the treatment of underlying assets driven by more structured stochastic processes, are taken into consideration in [17] and [18].…”

Section: A Short Quantization Reviewmentioning

confidence: 99%

A Quantization Approach to the Counterparty Credit Exposure Estimation

et al. 2015

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During recent years the counterparty risk subject has received a growing attention because of the so called Basel Accord. In particular the Basel III Accord asks the banks to fulfill finer conditions concerning counterparty credit exposures arising from banks' derivatives, securities financing transactions, default and downgrade risks characterizing the Over The Counter (OTC) derivatives market, etc. Consequently the development of effective and more accurate measures of risk have been pushed, particularly focusing on the estimate of the future fair value of derivatives with respect to prescribed time horizon and fixed grid of time buckets . Standard methods used to treat the latter scenario are mainly based on ad hoc implementations of the classic Monte Carlo (MC) approach, which is characterized by a high computational time, strongly dependent on the number of considered assets. This is why many financial players moved to more enhanced Technologies, e.g., grid computing and Graphics Processing Units (GPUs) capabilities. In this paper we show how to implement the quantization technique, in order to accurately estimate both pricing and volatility values. Our approach is tested to produce effective results for the counterparty risk evaluation, with a big improvement concerning required time to run when compared to MC approach.

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“…The first two steps have been already solved in [12]. We discuss in-depth the third one in Section 2.2).…”

Section: Introductionmentioning

confidence: 99%

Sharp Rate for the Dual Quantization Problem

Pagès¹,

Wilbertz²

2018

Séminaire De Probabilités XLIX

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In this paper we establish the sharp rate of the optimal dual quantization problem. The notion of dual quantization was recently introduced in [12], where it has been shown that, at least in a Euclidean setting, dual quantizers are based on a Delaunay triangulation, the dual counterpart of the Voronoi tessellation on which "regular" quantization relies. Moreover, this new approach shares an intrinsic stationarity property, which makes it very valuable for numerical applications.We establish in this paper the counterpart for dual quantization of the celebrated Zador theorem, which describes the sharp asymptotics for the quantization error when the quantizer size tends to infinity. On the way we establish an extension of the so-called Pierce Lemma by a random quantization argument. Numerical results confirm our choices.

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Intrinsic Stationarity for Vector Quantization: Foundation of Dual Quantization

Cited by 17 publications

References 18 publications

Introduction to vector quantization and its applications for numerics

Introduction to vector quantization and its applications for numerics

A Quantization Approach to the Counterparty Credit Exposure Estimation

Sharp Rate for the Dual Quantization Problem

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