47 research outputs found
Functional quantization and metric entropy for Riemann-Liouville processes
We derive a high-resolution formula for the -quantization errors of
Riemann-Liouville processes and the sharp Kolmogorov entropy asymptotics for
related Sobolev balls. We describe a quantization procedure which leads to
asymptotically optimal functional quantizers. Regular variation of the
eigenvalues of the covariance operator plays a crucial role
High-resolution product quantization for Gaussian processes under sup-norm distortion
We derive high-resolution upper bounds for optimal product quantization of
pathwise contionuous Gaussian processes respective to the supremum norm on
[0,T]^d. Moreover, we describe a product quantization design which attains this
bound. This is achieved under very general assumptions on random series
expansions of the process. It turns out that product quantization is
asymptotically only slightly worse than optimal functional quantization. The
results are applied e.g. to fractional Brownian sheets and the
Ornstein-Uhlenbeck process.Comment: Version publi\'ee dans la revue Bernoulli, 13(3), 653-67
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
Functional quantization rate and mean regularity of processes with an application to L\'evy processes
We investigate the connections between the mean pathwise regularity of
stochastic processes and their L^r(P)-functional quantization rates as random
variables taking values in some L^p([0,T],dt)-spaces (0 < p <= r). Our main
tool is the Haar basis. We then emphasize that the derived functional
quantization rate may be optimal (e.g., for Brownian motion or symmetric stable
processes) so that the rate is optimal as a universal upper bound. As a first
application, we establish the O((log N)^{-1/2}) upper bound for general It\^o
processes which include multidimensional diffusions. Then, we focus on the
specific family of L\'evy processes for which we derive a general quantization
rate based on the regular variation properties of its L\'evy measure at 0. The
case of compound Poisson processes, which appear as degenerate in the former
approach, is studied specifically: we observe some rates which are between the
finite-dimensional and infinite-dimensional ``usual'' ratesComment: 43p., issued in Annals of Applied Probability, 18(2):427-46
The local quantization behavior of absolutely continuous probabilities
For a large class of absolutely continuous probabilities it is shown
that, for , for -optimal -codebooks , and any Voronoi
partition with respect to the local probabilities
satisfy while the local
-quantization errors satisfy as long as the partition sets intersect a fixed compact
set in the interior of the support of .Comment: Published in at http://dx.doi.org/10.1214/11-AOP663 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Critical dimension for quadratic functional quantization
16 pagesIn this paper we tackle the asymptotics of the critical dimension for quadratic functional quantization of Gaussian stochastic processes as the quantization level goes to infinity, the smallest dimensional truncation of an optimal quantization of the process which is ''fully" quantized. We first establish a lower bound for this critical dimension based on the regular variation index of the eigenvalues of the Karhunen-Loève expansion of the process. This lower bound is consistent with the commonly shared sharp rate conjecture (and supported by extensive numerical experiments). Moreover, we show that, conversely, constructive optimized quadratic functional quantizations based on this critical dimension rate are always asymptotically optimal (strong admissibility result)
Expansions for Gaussian processes and Parseval frames
We derive a precise link between series expansions of Gaussian random vectors
in a Banach space and Parseval frames in their reproducing kernel Hilbert
space. The results are applied to pathwise continuous Gaussian processes and a
new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived.
In the end an extension of this result to Gaussian stationary processes with
convex covariance function is established.Comment: 20 page