4,371 research outputs found
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
Logarithmic and Riesz Equilibrium for Multiple Sources on the Sphere --- the Exceptional Case
We consider the minimal discrete and continuous energy problems on the unit
sphere in the Euclidean space in the presence
of an external field due to finitely many localized charge distributions on
, where the energy arises from the Riesz potential (
is the Euclidean distance) for the critical Riesz parameter if and the logarithmic potential if . Individually, a
localized charge distribution is either a point charge or assumed to be
rotationally symmetric. The extremal measure solving the continuous external
field problem for weak fields is shown to be the uniform measure on the sphere
but restricted to the exterior of spherical caps surrounding the localized
charge distributions. The radii are determined by the relative strengths of the
generating charges. Furthermore, we show that the minimal energy points solving
the related discrete external field problem are confined to this support. For
, we show that for point sources on the sphere, the equilibrium
measure has support in the complement of the union of specified spherical caps
about the sources. Numerical examples are provided to illustrate our results.Comment: 23 pages, 4 figure
Population extremal optimisation for discrete multi-objective optimisation problems
The power to solve intractable optimisation problems is often found through population based evolutionary methods. These include, but are not limited to, genetic algorithms, particle swarm optimisation, differential evolution and ant colony optimisation. While showing much promise as an effective optimiser, extremal optimisation uses only a single solution in its canonical form – and there are no standard population mechanics. In this paper, two population models for extremal optimisation are proposed and applied to a multi-objective version of the generalised assignment problem. These models use novel intervention/interaction strategies as well as collective memory in order to allow individual population members to work together. Additionally, a general non-dominated local search algorithm is developed and tested. Overall, the results show that improved attainment surfaces can be produced using population based interactions over not using them. The new EO approach is also shown to be highly competitive with an implementation of NSGA-II.No Full Tex
Utilization and Maintenance in a Model with Terminal Scrapping
We draw on three strands of literature dealing with utilization, maintenance, and scrapping in order to analyze the properties of the respective policies and their interac-tions. We do so by focusing on the last period of the received multi-period service life model and extending it in three directions: first, by associating the physical deteriora-tion of equipment to the intensity of its utilization and maintenance; second, by ex-panding on the range of explainable operating policies to allow for idling, mothballing, capacity depleting, capacity preserving, full capacity, upgrading, and downgrading; and, third, by linking the operating policies to the capital policy of scrapping. Owing to these enhancements, the analysis leads to several important findings. One among them is that optimal operating policies behave usually in opposite directions, proceed-ing in time from harder to softer or vice versa, depending on the net revenue earning capability of the equipment under consideration. Another is that profit (loss) making equipment is scrappable iff on the average the operating capital deteriorates faster (slower), or equivalently improves slower (faster), than the scrapping capital. And still an-other result is that operating policies are determined jointly with scrapping policy capi-tal policies, thus suggesting that empirical investigations of their determinants should allow for this simultaneityUtilization, maintenance, idling, mothballing, capacity depleting, capacity pre-serving, upgrading, downgrading, scrapping
Large deviation asymptotics for occupancy problems
In the standard formulation of the occupancy problem one considers the
distribution of r balls in n cells, with each ball assigned independently to a
given cell with probability 1/n. Although closed form expressions can be given
for the distribution of various interesting quantities (such as the fraction of
cells that contain a given number of balls), these expressions are often of
limited practical use. Approximations provide an attractive alternative, and in
the present paper we consider a large deviation approximation as r and n tend
to infinity. In order to analyze the problem we first consider a dynamical
model, where the balls are placed in the cells sequentially and ``time''
corresponds to the number of balls that have already been thrown. A complete
large deviation analysis of this ``process level'' problem is carried out, and
the rate function for the original problem is then obtained via the contraction
principle. The variational problem that characterizes this rate function is
analyzed, and a fairly complete and explicit solution is obtained. The
minimizing trajectories and minimal cost are identified up to two constants,
and the constants are characterized as the unique solution to an elementary
fixed point problem. These results are then used to solve a number of
interesting problems, including an overflow problem and the partial coupon
collector's problem.Comment: Published by the Institute of Mathematical Statistics
(http://www.imstat.org) in the Annals of Probability
(http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/00911790400000013
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