676 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
Greedy vector quantization
We investigate the greedy version of the -optimal vector quantization
problem for an -valued random vector . We show the
existence of a sequence such that minimizes
(-mean quantization error at level induced by
). We show that this sequence produces -rate
optimal -tuples ( the -mean
quantization error at level induced by goes to at rate
). Greedy optimal sequences also satisfy, under natural
additional assumptions, the distortion mismatch property: the -tuples
remain rate optimal with respect to the -norms, .
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 ) or stochastic
versions.Comment: 31 pages, 4 figures, few typos corrected (now an extended version of
an eponym paper to appear in Journal of Approximation
Distributed Functional Scalar Quantization Simplified
Distributed functional scalar quantization (DFSQ) theory provides optimality
conditions and predicts performance of data acquisition systems in which a
computation on acquired data is desired. We address two limitations of previous
works: prohibitively expensive decoder design and a restriction to sources with
bounded distributions. We rigorously show that a much simpler decoder has
equivalent asymptotic performance as the conditional expectation estimator
previously explored, thus reducing decoder design complexity. The simpler
decoder has the feature of decoupled communication and computation blocks.
Moreover, we extend the DFSQ framework with the simpler decoder to acquire
sources with infinite-support distributions such as Gaussian or exponential
distributions. Finally, through simulation results we demonstrate that
performance at moderate coding rates is well predicted by the asymptotic
analysis, and we give new insight on the rate of convergence
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