2,193 research outputs found
A vector quantization approach to universal noiseless coding and quantization
A two-stage code is a block code in which each block of data is coded in two stages: the first stage codes the identity of a block code among a collection of codes, and the second stage codes the data using the identified code. The collection of codes may be noiseless codes, fixed-rate quantizers, or variable-rate quantizers. We take a vector quantization approach to two-stage coding, in which the first stage code can be regarded as a vector quantizer that “quantizes” the input data of length n to one of a fixed collection of block codes. We apply the generalized Lloyd algorithm to the first-stage quantizer, using induced measures of rate and distortion, to design locally optimal two-stage codes. On a source of medical images, two-stage variable-rate vector quantizers designed in this way outperform standard (one-stage) fixed-rate vector quantizers by over 9 dB. The tail of the operational distortion-rate function of the first-stage quantizer determines the optimal rate of convergence of the redundancy of a universal sequence of two-stage codes. We show that there exist two-stage universal noiseless codes, fixed-rate quantizers, and variable-rate quantizers whose per-letter rate and distortion redundancies converge to zero as (k/2)n -1 log n, when the universe of sources has finite dimension k. This extends the achievability part of Rissanen's theorem from universal noiseless codes to universal quantizers. Further, we show that the redundancies converge as O(n-1) when the universe of sources is countable, and as O(n-1+ϵ) when the universe of sources is infinite-dimensional, under appropriate conditions
Robust Independent Component Analysis via Minimum Divergence Estimation
Independent component analysis (ICA) has been shown to be useful in many
applications. However, most ICA methods are sensitive to data contamination and
outliers. In this article we introduce a general minimum U-divergence framework
for ICA, which covers some standard ICA methods as special cases. Within the
U-family we further focus on the gamma-divergence due to its desirable property
of super robustness, which gives the proposed method gamma-ICA. Statistical
properties and technical conditions for the consistency of gamma-ICA are
rigorously studied. In the limiting case, it leads to a necessary and
sufficient condition for the consistency of MLE-ICA. This necessary and
sufficient condition is weaker than the condition known in the literature.
Since the parameter of interest in ICA is an orthogonal matrix, a geometrical
algorithm based on gradient flows on special orthogonal group is introduced to
implement gamma-ICA. Furthermore, a data-driven selection for the gamma value,
which is critical to the achievement of gamma-ICA, is developed. The
performance, especially the robustness, of gamma-ICA in comparison with
standard ICA methods is demonstrated through experimental studies using
simulated data and image data.Comment: 7 figure
Equation-free implementation of statistical moment closures
We present a general numerical scheme for the practical implementation of
statistical moment closures suitable for modeling complex, large-scale,
nonlinear systems. Building on recently developed equation-free methods, this
approach numerically integrates the closure dynamics, the equations of which
may not even be available in closed form. Although closure dynamics introduce
statistical assumptions of unknown validity, they can have significant
computational advantages as they typically have fewer degrees of freedom and
may be much less stiff than the original detailed model. The closure method can
in principle be applied to a wide class of nonlinear problems, including
strongly-coupled systems (either deterministic or stochastic) for which there
may be no scale separation. We demonstrate the equation-free approach for
implementing entropy-based Eyink-Levermore closures on a nonlinear stochastic
partial differential equation.Comment: 7 pages, 2 figure
Quasi-concave density estimation
Maximum likelihood estimation of a log-concave probability density is
formulated as a convex optimization problem and shown to have an equivalent
dual formulation as a constrained maximum Shannon entropy problem. Closely
related maximum Renyi entropy estimators that impose weaker concavity
restrictions on the fitted density are also considered, notably a minimum
Hellinger discrepancy estimator that constrains the reciprocal of the
square-root of the density to be concave. A limiting form of these estimators
constrains solutions to the class of quasi-concave densities.Comment: Published in at http://dx.doi.org/10.1214/10-AOS814 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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