Pointwise Convergence of the Lloyd I  Algorithm in Higher Dimension

Pagès, Gilles; Yu, Jun

doi:10.1137/151005622

Cited by 17 publications

(17 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For more details about these original stochastic optimization procedures, mostly devised in the 1950's, we refer e.g. to [5,35] for CLV Q and [25,18,39] for (randomized) Lloyd's I procedure or more applied textbooks like [20]. These procedures have been extensively implemented to compute for numerical probability purposes optimal grids of d-dimensional normal distributions N (0, I d ) for d = 1, .…”

Section: The Multidimensional Quadratic Case (Higher Dimensions)mentioning

confidence: 99%

Greedy vector quantization

Luschgy

Pagès

2015

Journal of Approximation Theory

Self Cite

View full text Add to dashboard Cite

We investigate the greedy version of the L p -optimal vector quantization problem for anby (a 1 , . . . , a N −1 , a)). We show that this sequence produces L p -rate optimal N -tuples a (N ) = (a 1 , . . . , a N ) (i.e. the L p -mean quantization error at level N induced by a (N ) goes to 0 at rate N − 1 d ). Greedy optimal sequences also satisfy, under natural additional assumptions, the distortion mismatch property: the N -tuples a (N ) remain rate optimal with respect to the L q -norms, p ≤ q < p + d. Finally, we propose optimization methods to compute greedy sequences, adapted from usual Lloyd's I and Competitive Learning Vector Quantization procedures, either in their deterministic (implementable when d = 1) or stochastic versions.

show abstract

Section: The Multidimensional Quadratic Case (Higher Dimensions)mentioning

confidence: 99%

Greedy vector quantization

Luschgy

Pagès

2015

Journal of Approximation Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…For numerical implementations, the search of stationary quantizers is based on zero search recursive procedures like Newton-Raphson algorithm for real valued random variables, and some algorithms like Lloyd's I algorithms (see e.g. [9,21]), the Competitive Learning Vector Quantization (CLVQ) algorithm (see [9]) or stochastic algorithms (see [18]) in the multidimensional framework. Optimal quantization grids associated to multivariate Gaussian random vectors can be downloaded on the website www.quantize.math-fi.com.…”

Section: Brief Background On Optimal Quantizationmentioning

confidence: 99%

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Fiorin

Pagès²,

Sagna

2018

Methodol Comput Appl Probab

Self Cite

View full text Add to dashboard Cite

We introduce a new approach to quantize the Euler scheme of an R d -valued diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of fast online quantization in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the k-th discretization step t k is a cumulative of the marginal quantization error up to time t k . Numerical experiments are performed for the pricing of a Basket call option, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations to show the performances of the method.

show abstract

“…Like for the CLV Q algorithm, the convergence results for the Lloyd procedure are still partial, even in its original (deterministic) form (3.43) (see e.g. [67]). …”

Section: Randomized Lloyd's Procedurementioning

confidence: 99%

“…to [8,59] for CLV Q, [25,40,67] for (randomized) Lloyd's procedure or more applied textbooks like [30]. 3.4.…”

Section: Randomized Lloyd's Procedurementioning

confidence: 99%

Introduction to vector quantization and its applications for numerics

Pagès

2015

ESAIM: Proc.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Pointwise Convergence of the Lloyd I Algorithm in Higher Dimension

Cited by 17 publications

References 31 publications

Greedy vector quantization

Greedy vector quantization

Product Markovian Quantization of a Diffusion Process with Applications to Finance

Introduction to vector quantization and its applications for numerics

Contact Info

Product

Resources

About