3,175 research outputs found
Approximation learning methods of Harmonic Mappings in relation to Hardy Spaces
A new Hardy space Hardy space approach of Dirichlet type problem based on
Tikhonov regularization and Reproducing Hilbert kernel space is discussed in
this paper, which turns out to be a typical extremal problem located on the
upper upper-high complex plane. If considering this in the Hardy space, the
optimization operator of this problem will be highly simplified and an
efficient algorithm is possible. This is mainly realized by the help of
reproducing properties of the functions in the Hardy space of upper-high
complex plane, and the detail algorithm is proposed. Moreover, harmonic
mappings, which is a significant geometric transformation, are commonly used in
many applications such as image processing, since it describes the energy
minimization mappings between individual manifolds. Particularly, when we focus
on the planer mappings between two Euclid planer regions, the harmonic mappings
are exist and unique, which is guaranteed solidly by the existence of harmonic
function. This property is attractive and simulation results are shown in this
paper to ensure the capability of applications such as planer shape distortion
and surface registration.Comment: 2016 3rd International Conference on Informative and Cybernetics for
Computational Social Systems (ICCSS
Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation
We investigate an encoding scheme for lossy compression of a binary symmetric
source based on simple spatially coupled Low-Density Generator-Matrix codes.
The degree of the check nodes is regular and the one of code-bits is Poisson
distributed with an average depending on the compression rate. The performance
of a low complexity Belief Propagation Guided Decimation algorithm is
excellent. The algorithmic rate-distortion curve approaches the optimal curve
of the ensemble as the width of the coupling window grows. Moreover, as the
check degree grows both curves approach the ultimate Shannon rate-distortion
limit. The Belief Propagation Guided Decimation encoder is based on the
posterior measure of a binary symmetric test-channel. This measure can be
interpreted as a random Gibbs measure at a "temperature" directly related to
the "noise level of the test-channel". We investigate the links between the
algorithmic performance of the Belief Propagation Guided Decimation encoder and
the phase diagram of this Gibbs measure. The phase diagram is investigated
thanks to the cavity method of spin glass theory which predicts a number of
phase transition thresholds. In particular the dynamical and condensation
"phase transition temperatures" (equivalently test-channel noise thresholds)
are computed. We observe that: (i) the dynamical temperature of the spatially
coupled construction saturates towards the condensation temperature; (ii) for
large degrees the condensation temperature approaches the temperature (i.e.
noise level) related to the information theoretic Shannon test-channel noise
parameter of rate-distortion theory. This provides heuristic insight into the
excellent performance of the Belief Propagation Guided Decimation algorithm.
The paper contains an introduction to the cavity method
Efficient sphere-covering and converse measure concentration via generalized coding theorems
Suppose A is a finite set equipped with a probability measure P and let M be
a ``mass'' function on A. We give a probabilistic characterization of the most
efficient way in which A^n can be almost-covered using spheres of a fixed
radius. An almost-covering is a subset C_n of A^n, such that the union of the
spheres centered at the points of C_n has probability close to one with respect
to the product measure P^n. An efficient covering is one with small mass
M^n(C_n); n is typically large. With different choices for M and the geometry
on A our results give various corollaries as special cases, including Shannon's
data compression theorem, a version of Stein's lemma (in hypothesis testing),
and a new converse to some measure concentration inequalities on discrete
spaces. Under mild conditions, we generalize our results to abstract spaces and
non-product measures.Comment: 29 pages. See also http://www.stat.purdue.edu/~yiannis
Bounds on inference
Lower bounds for the average probability of error of estimating a hidden
variable X given an observation of a correlated random variable Y, and Fano's
inequality in particular, play a central role in information theory. In this
paper, we present a lower bound for the average estimation error based on the
marginal distribution of X and the principal inertias of the joint distribution
matrix of X and Y. Furthermore, we discuss an information measure based on the
sum of the largest principal inertias, called k-correlation, which generalizes
maximal correlation. We show that k-correlation satisfies the Data Processing
Inequality and is convex in the conditional distribution of Y given X. Finally,
we investigate how to answer a fundamental question in inference and privacy:
given an observation Y, can we estimate a function f(X) of the hidden random
variable X with an average error below a certain threshold? We provide a
general method for answering this question using an approach based on
rate-distortion theory.Comment: Allerton 2013 with extended proof, 10 page
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