97 research outputs found
Quadratic optimal functional quantization of stochastic processes and numerical applications
In this paper, we present an overview of the recent developments of
functional quantization of stochastic processes, with an emphasis on the
quadratic case. Functional quantization is a way to approximate a process,
viewed as a Hilbert-valued random variable, using a nearest neighbour
projection on a finite codebook. A special emphasis is made on the
computational aspects and the numerical applications, in particular the pricing
of some path-dependent European options.Comment: 41 page
High-Rate Vector Quantization for the Neyman-Pearson Detection of Correlated Processes
This paper investigates the effect of quantization on the performance of the
Neyman-Pearson test. It is assumed that a sensing unit observes samples of a
correlated stationary ergodic multivariate process. Each sample is passed
through an N-point quantizer and transmitted to a decision device which
performs a binary hypothesis test. For any false alarm level, it is shown that
the miss probability of the Neyman-Pearson test converges to zero exponentially
as the number of samples tends to infinity, assuming that the observed process
satisfies certain mixing conditions. The main contribution of this paper is to
provide a compact closed-form expression of the error exponent in the high-rate
regime i.e., when the number N of quantization levels tends to infinity,
generalizing previous results of Gupta and Hero to the case of non-independent
observations. If d represents the dimension of one sample, it is proved that
the error exponent converges at rate N^{2/d} to the one obtained in the absence
of quantization. As an application, relevant high-rate quantization strategies
which lead to a large error exponent are determined. Numerical results indicate
that the proposed quantization rule can yield better performance than existing
ones in terms of detection error.Comment: 47 pages, 7 figures, 1 table. To appear in the IEEE Transactions on
Information Theor
Quantization as histogram segmentation: globally optimal scalar quantizer design in network systems
We propose a polynomial-time algorithm for optimal scalar quantizer design on discrete-alphabet sources. Special cases of the proposed approach yield optimal design algorithms for fixed-rate and entropy-constrained scalar quantizers, multi-resolution scalar quantizers, multiple description scalar quantizers, and Wyner-Ziv scalar quantizers. The algorithm guarantees globally optimal solutions for fixed-rate and entropy-constrained scalar quantizers and constrained optima for the other coding scenarios. We derive the algorithm by demonstrating the connection between scalar quantization, histogram segmentation, and the shortest path problem in a certain directed acyclic graph
On the effect of quantization on performance at high rates
We study the effect of quantization on the performance of a scalar dynamical system in the high rate regime. We evaluate the LQ cost for two commonly used quantizers: uniform and logarithmic and provide a lower bound on performance of any centroid-based quantizer based on entropy arguments. We also consider the case when the channel drops data packets stochastically
Quantization as Histogram Segmentation: Optimal Scalar Quantizer Design in Network Systems
An algorithm for scalar quantizer design on discrete-alphabet sources is proposed. The proposed algorithm can be used to design fixed-rate and entropy-constrained conventional scalar quantizers, multiresolution scalar quantizers, multiple description scalar quantizers, and Wyner–Ziv scalar quantizers. The algorithm guarantees globally optimal solutions for conventional fixed-rate scalar quantizers and entropy-constrained scalar quantizers. For the other coding scenarios, the algorithm yields the best code among all codes that meet a given convexity constraint. In all cases, the algorithm run-time is polynomial in the size of the source alphabet. The algorithm derivation arises from a demonstration of the connection between scalar quantization, histogram segmentation, and the shortest path problem in a certain directed acyclic graph
Introduction to vector quantization and its applications for numerics*
Proceedings of CEMRACS 2013 - Modelling and simulation of complex systems: stochastic and deterministic approaches. : T. Lelièvre et al. EditorsInternational audienceWe 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 ℝd (equipped with the canonical Euclidean norm) and the learning procedures that allow to design optimal quantizers (CLVQ 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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