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math/0509325
|
On $Z_{2^k}$-Dual Binary Codes
|
math.CO cs.IT math.IT
|
A new generalization of the Gray map is introduced. The new generalization
$\Phi: Z_{2^k}^n \to Z_{2}^{2^{k-1}n}$ is connected with the known generalized
Gray map $\phi$ in the following way: if we take two dual linear
$Z_{2^k}$-codes and construct binary codes from them using the generalizations
$\phi$ and $\Phi$ of the Gray map, then the weight enumerators of the binary
codes obtained will satisfy the MacWilliams identity. The classes of
$Z_{2^k}$-linear Hadamard codes and co-$Z_{2^k}$-linear extended 1-perfect
codes are described, where co-$Z_{2^k}$-linearity means that the code can be
obtained from a linear $Z_{2^k}$-code with the help of the new generalized Gray
map. Keywords: Gray map, Hadamard codes, MacWilliams identity, perfect codes,
$Z_{2^k}$-linearity
|
math/0509358
|
On decomposability of 4-ary distance 2 MDS codes, double-codes, and
n-quasigroups of order 4
|
math.CO cs.IT math.IT
|
A subset $S$ of $\{0,1,...,2t-1\}^n$ is called a $t$-fold MDS code if every
line in each of $n$ base directions contains exactly $t$ elements of $S$. The
adjacency graph of a $t$-fold MDS code is not connected if and only if the
characteristic function of the code is the repetition-free sum of the
characteristic functions of $t$-fold MDS codes of smaller lengths.
In the case $t=2$, the theory has the following application. The union of two
disjoint $(n,4^{n-1},2)$ MDS codes in $\{0,1,2,3\}^n$ is a double-MDS-code. If
the adjacency graph of the double-MDS-code is not connected, then the
double-code can be decomposed into double-MDS-codes of smaller lengths. If the
graph has more than two connected components, then the MDS codes are also
decomposable. The result has an interpretation as a test for reducibility of
$n$-quasigroups of order 4. Keywords: MDS codes, n-quasigroups,
decomposability, reducibility, frequency hypercubes, latin hypercubes
|
math/0509575
|
Evolutionary Trees and the Ising Model on the Bethe Lattice: a Proof of
Steel's Conjecture
|
math.PR cs.CE cs.DS math.CA math.CO math.ST q-bio.PE stat.TH
|
A major task of evolutionary biology is the reconstruction of phylogenetic
trees from molecular data. The evolutionary model is given by a Markov chain on
a tree. Given samples from the leaves of the Markov chain, the goal is to
reconstruct the leaf-labelled tree.
It is well known that in order to reconstruct a tree on $n$ leaves, sample
sequences of length $\Omega(\log n)$ are needed. It was conjectured by M. Steel
that for the CFN/Ising evolutionary model, if the mutation probability on all
edges of the tree is less than $p^{\ast} = (\sqrt{2}-1)/2^{3/2}$, then the tree
can be recovered from sequences of length $O(\log n)$. The value $p^{\ast}$ is
given by the transition point for the extremality of the free Gibbs measure for
the Ising model on the binary tree. Steel's conjecture was proven by the second
author in the special case where the tree is "balanced." The second author also
proved that if all edges have mutation probability larger than $p^{\ast}$ then
the length needed is $n^{\Omega(1)}$. Here we show that Steel's conjecture
holds true for general trees by giving a reconstruction algorithm that recovers
the tree from $O(\log n)$-length sequences when the mutation probabilities are
discretized and less than $p^\ast$. Our proof and results demonstrate that
extremality of the free Gibbs measure on the infinite binary tree, which has
been studied before in probability, statistical physics and computer science,
determines how distinguishable are Gibbs measures on finite binary trees.
|
math/0509620
|
On diameter perfect constant-weight ternary codes
|
math.CO cs.IT math.IT
|
From cosets of binary Hamming codes we construct diameter perfect
constant-weight ternary codes with weight $n-1$ (where $n$ is the code length)
and distances 3 and 5. The class of distance 5 codes has parameters unknown
before. Keywords: constant-weight codes, ternary codes, perfect codes, diameter
perfect codes, perfect matchings, Preparata codes
|
math/0510276
|
An algorithmic and a geometric characterization of Coarsening At Random
|
math.ST cs.AI stat.ME stat.TH
|
We show that the class of conditional distributions satisfying the coarsening
at Random (CAR) property for discrete data has a simple and robust algorithmic
description based on randomized uniform multicovers: combinatorial objects
generalizing the notion of partition of a set. However, the complexity of a
given CAR mechanism can be large: the maximal "height" of the needed
multicovers can be exponential in the number of points in the sample space. The
results stem from a geometric interpretation of the set of CAR distributions as
a convex polytope and a characterization of its extreme points. The hierarchy
of CAR models defined in this way could be useful in parsimonious statistical
modelling of CAR mechanisms, though the results also raise doubts in applied
work as to the meaningfulness of the CAR assumption in its full generality.
|
math/0510521
|
On surrogate loss functions and $f$-divergences
|
math.ST cs.IT math.IT stat.TH
|
The goal of binary classification is to estimate a discriminant function
$\gamma$ from observations of covariate vectors and corresponding binary
labels. We consider an elaboration of this problem in which the covariates are
not available directly but are transformed by a dimensionality-reducing
quantizer $Q$. We present conditions on loss functions such that empirical risk
minimization yields Bayes consistency when both the discriminant function and
the quantizer are estimated. These conditions are stated in terms of a general
correspondence between loss functions and a class of functionals known as
Ali-Silvey or $f$-divergence functionals. Whereas this correspondence was
established by Blackwell [Proc. 2nd Berkeley Symp. Probab. Statist. 1 (1951)
93--102. Univ. California Press, Berkeley] for the 0--1 loss, we extend the
correspondence to the broader class of surrogate loss functions that play a key
role in the general theory of Bayes consistency for binary classification. Our
result makes it possible to pick out the (strict) subset of surrogate loss
functions that yield Bayes consistency for joint estimation of the discriminant
function and the quantizer.
|
math/0512263
|
Metric and probabilistic information associated with Fredholm integral
equations of the first kind
|
math.CA cs.IT math.IT
|
The problem of evaluating the information associated with Fredholm integral
equations of the first kind, when the integral operator is self-adjoint and
compact, is considered here. The data function is assumed to be perturbed
gently by an additive noise so that it still belongs to the range of the
operator. First we estimate upper and lower bounds for the epsilon-capacity
(and then for the metric information), and explicit computations in some
specific cases are given; then the problem is reformulated from a probabilistic
viewpoint and use is made of the probabilistic information theory. The results
obtained by these two approaches are then compared.
|
math/0602070
|
The Matrix-Forest Theorem and Measuring Relations in Small Social Groups
|
math.CO cs.IR math.AG
|
We propose a family of graph structural indices related to the Matrix-forest
theorem. The properties of the basic index that expresses the mutual
connectivity of two vertices are studied in detail. The derivative indices that
measure "dissociation," "solitariness," and "provinciality" of vertices are
also considered. A nonstandard metric on the set of vertices is introduced,
which is determined by their connectivity. The application of these indices in
sociometry is discussed.
|
math/0602171
|
Preference fusion when the number of alternatives exceeds two: indirect
scoring procedures
|
math.OC cs.MA math.CO
|
We consider the problem of aggregation of incomplete preferences represented
by arbitrary binary relations or incomplete paired comparison matrices. For a
number of indirect scoring procedures we examine whether or not they satisfy
the axiom of self-consistent monotonicity. The class of {\em win-loss combining
scoring procedures} is introduced which contains a majority of known scoring
procedures. Two main results are established. According to the first one, every
win-loss combining scoring procedure breaks self-consistent monotonicity. The
second result provides a sufficient condition of satisfying self-consistent
monotonicity.
|
math/0602505
|
MDL Convergence Speed for Bernoulli Sequences
|
math.ST cs.IT cs.LG math.IT math.PR stat.TH
|
The Minimum Description Length principle for online sequence
estimation/prediction in a proper learning setup is studied. If the underlying
model class is discrete, then the total expected square loss is a particularly
interesting performance measure: (a) this quantity is finitely bounded,
implying convergence with probability one, and (b) it additionally specifies
the convergence speed. For MDL, in general one can only have loss bounds which
are finite but exponentially larger than those for Bayes mixtures. We show that
this is even the case if the model class contains only Bernoulli distributions.
We derive a new upper bound on the prediction error for countable Bernoulli
classes. This implies a small bound (comparable to the one for Bayes mixtures)
for certain important model classes. We discuss the application to Machine
Learning tasks such as classification and hypothesis testing, and
generalization to countable classes of i.i.d. models.
|
math/0602522
|
Characterizations of scoring methods for preference aggregation
|
math.OC cs.MA math.FA
|
The paper surveys more than forty characterizations of scoring methods for
preference aggregation and contains one new result. A general scoring operator
is {\it self-consistent} if alternative $i$ is assigned a greater score than
$j$ whenever $i$ gets no worse (better) results of comparisons and its
`opponents' are assigned respectively greater (no smaller) scores than those of
$j$. We prove that self-consistency is satisfied if and only if the application
of a scoring operator reduces to the solution of a homogeneous system of
algebraic equations with a monotone function on the left-hand side.
|
math/0602552
|
From Incomplete Preferences to Ranking via Optimization
|
math.OC cs.MA math.CO
|
We consider methods for aggregating preferences that are based on the
resolution of discrete optimization problems. The preferences are represented
by arbitrary binary relations (possibly weighted) or incomplete paired
comparison matrices. This incomplete case remains practically unexplored so
far. We examine the properties of several known methods and propose one new
method. In particular, we test whether these methods obey a new axiom referred
to as {\it Self-Consistent Monotonicity}. Some results are established that
characterize solutions of the related optimization problems.
|
math/0603155
|
Vers une commande multivariable sans mod\`ele
|
math.OC cs.CE cs.RO physics.class-ph
|
A control strategy without any precise mathematical model is derived for
linear or nonlinear systems which are assumed to be finite-dimensional. Two
convincing numerical simulations are provided.
|
math/0604233
|
Generalization error bounds in semi-supervised classification under the
cluster assumption
|
math.ST cs.LG stat.TH
|
We consider semi-supervised classification when part of the available data is
unlabeled. These unlabeled data can be useful for the classification problem
when we make an assumption relating the behavior of the regression function to
that of the marginal distribution. Seeger (2000) proposed the well-known
"cluster assumption" as a reasonable one. We propose a mathematical formulation
of this assumption and a method based on density level sets estimation that
takes advantage of it to achieve fast rates of convergence both in the number
of unlabeled examples and the number of labeled examples.
|
math/0605498
|
Cross-Entropic Learning of a Machine for the Decision in a Partially
Observable Universe
|
math.OC cs.AI cs.LG cs.NE cs.RO math.ST stat.TH
|
Revision of the paper previously entitled "Learning a Machine for the
Decision in a Partially Observable Markov Universe" In this paper, we are
interested in optimal decisions in a partially observable universe. Our
approach is to directly approximate an optimal strategic tree depending on the
observation. This approximation is made by means of a parameterized
probabilistic law. A particular family of hidden Markov models, with input
\emph{and} output, is considered as a model of policy. A method for optimizing
the parameters of these HMMs is proposed and applied. This optimization is
based on the cross-entropic principle for rare events simulation developed by
Rubinstein.
|
math/0605740
|
Sharp thresholds for high-dimensional and noisy recovery of sparsity
|
math.ST cs.IT math.IT stat.TH
|
The problem of consistently estimating the sparsity pattern of a vector
$\betastar \in \real^\mdim$ based on observations contaminated by noise arises
in various contexts, including subset selection in regression, structure
estimation in graphical models, sparse approximation, and signal denoising. We
analyze the behavior of $\ell_1$-constrained quadratic programming (QP), also
referred to as the Lasso, for recovering the sparsity pattern. Our main result
is to establish a sharp relation between the problem dimension $\mdim$, the
number $\spindex$ of non-zero elements in $\betastar$, and the number of
observations $\numobs$ that are required for reliable recovery. For a broad
class of Gaussian ensembles satisfying mutual incoherence conditions, we
establish existence and compute explicit values of thresholds $\ThreshLow$ and
$\ThreshUp$ with the following properties: for any $\epsilon > 0$, if $\numobs
> 2 (\ThreshUp + \epsilon) \log (\mdim - \spindex) + \spindex + 1$, then the
Lasso succeeds in recovering the sparsity pattern with probability converging
to one for large problems, whereas for $\numobs < 2 (\ThreshLow - \epsilon)
\log (\mdim - \spindex) + \spindex + 1$, then the probability of successful
recovery converges to zero. For the special case of the uniform Gaussian
ensemble, we show that $\ThreshLow = \ThreshUp = 1$, so that the threshold is
sharp and exactly determined.
|
math/0606315
|
Bayesian Regression of Piecewise Constant Functions
|
math.ST cs.LG math.PR stat.TH
|
We derive an exact and efficient Bayesian regression algorithm for piecewise
constant functions of unknown segment number, boundary location, and levels. It
works for any noise and segment level prior, e.g. Cauchy which can handle
outliers. We derive simple but good estimates for the in-segment variance. We
also propose a Bayesian regression curve as a better way of smoothing data
without blurring boundaries. The Bayesian approach also allows straightforward
determination of the evidence, break probabilities and error estimates, useful
for model selection and significance and robustness studies. We discuss the
performance on synthetic and real-world examples. Many possible extensions will
be discussed.
|
math/0606643
|
Entropy And Vision
|
math.PR cs.CV cs.DB cs.DM cs.LG math.CO
|
In vector quantization the number of vectors used to construct the codebook
is always an undefined problem, there is always a compromise between the number
of vectors and the quantity of information lost during the compression. In this
text we present a minimum of Entropy principle that gives solution to this
compromise and represents an Entropy point of view of signal compression in
general. Also we present a new adaptive Object Quantization technique that is
the same for the compression and the perception.
|
math/0606734
|
Codes in spherical caps
|
math.MG cs.IT math.IT
|
We consider bounds on codes in spherical caps and related problems in
geometry and coding theory. An extension of the Delsarte method is presented
that relates upper bounds on the size of spherical codes to upper bounds on
codes in caps. Several new upper bounds on codes in caps are derived.
Applications of these bounds to estimates of the kissing numbers and one-sided
kissing numbers are considered.
It is proved that the maximum size of codes in spherical caps for large
dimensions is determined by the maximum size of spherical codes, so these
problems are asymptotically equivalent.
|
math/0607243
|
An active curve approach for tomographic reconstruction of binary
radially symmetric objects
|
math.OC cs.CV
|
This paper deals with a method of tomographic reconstruction of radially
symmetric objects from a single radiograph, in order to study the behavior of
shocked material. The usual tomographic reconstruction algorithms such as
generalized inverse or filtered back-projection cannot be applied here because
data are very noisy and the inverse problem associated to single view
tomographic reconstruction is highly unstable. In order to improve the
reconstruction, we propose here to add some a priori assumptions on the looked
after object. One of these assumptions is that the object is binary and
consequently, the object may be described by the curves that separate the two
materials. We present a model that lives in BV space and leads to a non local
Hamilton-Jacobi equation, via a level set strategy. Numerical experiments are
performed (using level sets methods) on synthetic objects.
|
math/0607507
|
In-Degree and PageRank of Web pages: Why do they follow similar power
laws?
|
math.PR cs.IR
|
The PageRank is a popularity measure designed by Google to rank Web pages.
Experiments confirm that the PageRank obeys a `power law' with the same
exponent as the In-Degree. This paper presents a novel mathematical model that
explains this phenomenon. The relation between the PageRank and In-Degree is
modelled through a stochastic equation, which is inspired by the original
definition of the PageRank, and is analogous to the well-known distributional
identity for the busy period in the M/G/1 queue. Further, we employ the theory
of regular variation and Tauberian theorems to analytically prove that the tail
behavior of the PageRank and the In-Degree differ only by a multiplicative
factor, for which we derive a closed-form expression. Our analytical results
are in good agreement with experimental data.
|
math/0607648
|
Singular Values and Eigenvalues of Tensors: A Variational Approach
|
math.SP cs.IR cs.NA math.NA math.OC
|
We propose a theory of eigenvalues, eigenvectors, singular values, and
singular vectors for tensors based on a constrained variational approach much
like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are
particularly useful in generalizing certain areas where the spectral theory of
matrices has traditionally played an important role. For illustration, we will
discuss a multilinear generalization of the Perron-Frobenius theorem.
|
math/0608522
|
Graph Laplacians and their convergence on random neighborhood graphs
|
math.ST cs.LG stat.TH
|
Given a sample from a probability measure with support on a submanifold in
Euclidean space one can construct a neighborhood graph which can be seen as an
approximation of the submanifold. The graph Laplacian of such a graph is used
in several machine learning methods like semi-supervised learning,
dimensionality reduction and clustering. In this paper we determine the
pointwise limit of three different graph Laplacians used in the literature as
the sample size increases and the neighborhood size approaches zero. We show
that for a uniform measure on the submanifold all graph Laplacians have the
same limit up to constants. However in the case of a non-uniform measure on the
submanifold only the so called random walk graph Laplacian converges to the
weighted Laplace-Beltrami operator.
|
math/0608556
|
On optimal quantization rules for some problems in sequential
decentralized detection
|
math.ST cs.IT math.IT stat.TH
|
We consider the design of systems for sequential decentralized detection, a
problem that entails several interdependent choices: the choice of a stopping
rule (specifying the sample size), a global decision function (a choice between
two competing hypotheses), and a set of quantization rules (the local decisions
on the basis of which the global decision is made). This paper addresses an
open problem of whether in the Bayesian formulation of sequential decentralized
detection, optimal local decision functions can be found within the class of
stationary rules. We develop an asymptotic approximation to the optimal cost of
stationary quantization rules and exploit this approximation to show that
stationary quantizers are not optimal in a broad class of settings. We also
consider the class of blockwise stationary quantizers, and show that
asymptotically optimal quantizers are likelihood-based threshold rules.
|
math/0608571
|
Intensional Models for the Theory of Types
|
math.LO cs.AI
|
In this paper we define intensional models for the classical theory of types,
thus arriving at an intensional type logic ITL. Intensional models generalize
Henkin's general models and have a natural definition. As a class they do not
validate the axiom of Extensionality. We give a cut-free sequent calculus for
type theory and show completeness of this calculus with respect to the class of
intensional models via a model existence theorem. After this we turn our
attention to applications. Firstly, it is argued that, since ITL is truly
intensional, it can be used to model ascriptions of propositional attitude
without predicting logical omniscience. In order to illustrate this a small
fragment of English is defined and provided with an ITL semantics. Secondly, it
is shown that ITL models contain certain objects that can be identified with
possible worlds. Essential elements of modal logic become available within
classical type theory once the axiom of Extensionality is given up.
|
math/0608713
|
Occam's hammer: a link between randomized learning and multiple testing
FDR control
|
math.ST cs.LG stat.TH
|
We establish a generic theoretical tool to construct probabilistic bounds for
algorithms where the output is a subset of objects from an initial pool of
candidates (or more generally, a probability distribution on said pool). This
general device, dubbed "Occam's hammer'', acts as a meta layer when a
probabilistic bound is already known on the objects of the pool taken
individually, and aims at controlling the proportion of the objects in the set
output not satisfying their individual bound. In this regard, it can be seen as
a non-trivial generalization of the "union bound with a prior'' ("Occam's
razor''), a familiar tool in learning theory. We give applications of this
principle to randomized classifiers (providing an interesting alternative
approach to PAC-Bayes bounds) and multiple testing (where it allows to retrieve
exactly and extend the so-called Benjamini-Yekutieli testing procedure).
|
math/0609461
|
Cross-Entropy method: convergence issues for extended implementation
|
math.OC cs.LG cs.NE math.ST stat.TH
|
The cross-entropy method (CE) developed by R. Rubinstein is an elegant
practical principle for simulating rare events. The method approximates the
probability of the rare event by means of a family of probabilistic models. The
method has been extended to optimization, by considering an optimal event as a
rare event. CE works rather good when dealing with deterministic function
optimization. Now, it appears that two conditions are needed for a good
convergence of the method. First, it is necessary to have a family of models
sufficiently flexible for discriminating the optimal events. Indirectly, it
appears also that the function to be optimized should be deterministic. The
purpose of this paper is to consider the case of partially discriminating model
family, and of stochastic functions. It will be shown on simple examples that
the CE could fail when relaxing these hypotheses. Alternative improvements of
the CE method are investigated and compared on random examples in order to
handle this issue.
|
math/0609562
|
On quadratic residue codes and hyperelliptic curves
|
math.CO cs.IT math.AG math.IT math.NT
|
A long standing problem has been to develop "good" binary linear codes to be
used for error-correction. This paper investigates in some detail an attack on
this problem using a connection between quadratic residue codes and
hyperelliptic curves. One question which coding theory is used to attack is:
Does there exist a c<2 such that, for all sufficiently large $p$ and all
subsets S of GF(p), we have |X_S(GF(p))| < cp?
|
math/0610184
|
Adaptive Poisson disorder problem
|
math.PR cs.IT math.IT math.ST stat.TH
|
We study the quickest detection problem of a sudden change in the arrival
rate of a Poisson process from a known value to an unknown and unobservable
value at an unknown and unobservable disorder time. Our objective is to design
an alarm time which is adapted to the history of the arrival process and
detects the disorder time as soon as possible. In previous solvable versions of
the Poisson disorder problem, the arrival rate after the disorder has been
assumed a known constant. In reality, however, we may at most have some prior
information about the likely values of the new arrival rate before the disorder
actually happens, and insufficient estimates of the new rate after the disorder
happens. Consequently, we assume in this paper that the new arrival rate after
the disorder is a random variable. The detection problem is shown to admit a
finite-dimensional Markovian sufficient statistic, if the new rate has a
discrete distribution with finitely many atoms. Furthermore, the detection
problem is cast as a discounted optimal stopping problem with running cost for
a finite-dimensional piecewise-deterministic Markov process. This optimal
stopping problem is studied in detail in the special case where the new arrival
rate has Bernoulli distribution. This is a nontrivial optimal stopping problem
for a two-dimensional piecewise-deterministic Markov process driven by the same
point process. Using a suitable single-jump operator, we solve it fully,
describe the analytic properties of the value function and the stopping region,
and present methods for their numerical calculation. We provide a concrete
example where the value function does not satisfy the smooth-fit principle on a
proper subset of the connected, continuously differentiable optimal stopping
boundary, whereas it does on the complement of this set.
|
math/0611422
|
Cartes auto-organis\'{e}es pour l'analyse exploratoire de donn\'{e}es et
la visualisation
|
math.ST cs.NE nlin.AO stat.TH
|
This paper shows how to use the Kohonen algorithm to represent
multidimensional data, by exploiting the self-organizing property. It is
possible to get such maps as well for quantitative variables as for qualitative
ones, or for a mixing of both. The contents of the paper come from various
works by SAMOS-MATISSE members, in particular by E. de Bodt, B. Girard, P.
Letr\'{e}my, S. Ibbou, P. Rousset. Most of the examples have been studied with
the computation routines written by Patrick Letr\'{e}my, with the language
IML-SAS, which are available on the WEB page http://samos.univ-paris1.fr.
|
math/0611433
|
Working times in atypical forms of employment: the special case of
part-time work
|
math.ST cs.NE stat.TH
|
In the present article, we attempt to devise a typology of forms of part-time
employment by applying a widely used neuronal methodology called Kohonen maps.
Starting out with data that we describe using category-specific variables, we
show how it is possible to represent observations and the modalities of the
variables that define them simultaneously, on a single map. This allows us to
ascertain, and to try to describe, the main categories of part-time employment.
|
math/0611937
|
Remarks on Inheritance Systems
|
math.LO cs.AI
|
We try a conceptual analysis of inheritance diagrams, first in abstract
terms, and then compare to "normality" and the "small/big sets" of preferential
and related reasoning. The main ideas are about nodes as truth values and
information sources, truth comparison by paths, accessibility or relevance of
information by paths, relative normality, and prototypical reasoning.
|
math/0612046
|
On the Submodularity of Influence in Social Networks
|
math.PR cs.GT cs.SI
|
We prove and extend a conjecture of Kempe, Kleinberg, and Tardos (KKT) on the
spread of influence in social networks. A social network can be represented by
a directed graph where the nodes are individuals and the edges indicate a form
of social relationship. A simple way to model the diffusion of ideas,
innovative behavior, or ``word-of-mouth'' effects on such a graph is to
consider an increasing process of ``infected'' (or active) nodes: each node
becomes infected once an activation function of the set of its infected
neighbors crosses a certain threshold value. Such a model was introduced by KKT
in \cite{KeKlTa:03,KeKlTa:05} where the authors also impose several natural
assumptions: the threshold values are (uniformly) random; and the activation
functions are monotone and submodular. For an initial set of active nodes $S$,
let $\sigma(S)$ denote the expected number of active nodes at termination. Here
we prove a conjecture of KKT: we show that the function $\sigma(S)$ is
submodular under the assumptions above. We prove the same result for the
expected value of any monotone, submodular function of the set of active nodes
at termination.
|
math/0612682
|
Convergence Speed in Distributed Consensus and Control
|
math.OC cs.SY
|
We study the convergence speed of distributed iterative algorithms for the
consensus and averaging problems, with emphasis on the latter. We first
consider the case of a fixed communication topology. We show that a simple
adaptation of a consensus algorithm leads to an averaging algorithm. We prove
lower bounds on the worst-case convergence time for various classes of linear,
time-invariant, distributed consensus methods, and provide an algorithm that
essentially matches those lower bounds. We then consider the case of a
time-varying topology, and provide a polynomial-time averaging algorithm.
|
math/0701131
|
Compressed Sensing and Redundant Dictionaries
|
math.PR cs.IT math.IT
|
This article extends the concept of compressed sensing to signals that are
not sparse in an orthonormal basis but rather in a redundant dictionary. It is
shown that a matrix, which is a composition of a random matrix of certain type
and a deterministic dictionary, has small restricted isometry constants. Thus,
signals that are sparse with respect to the dictionary can be recovered via
Basis Pursuit from a small number of random measurements. Further, thresholding
is investigated as recovery algorithm for compressed sensing and conditions are
provided that guarantee reconstruction with high probability. The different
schemes are compared by numerical experiments.
|
math/0701142
|
On the use of self-organizing maps to accelerate vector quantization
|
math.ST cs.NE stat.TH
|
Self-organizing maps (SOM) are widely used for their topology preservation
property: neighboring input vectors are quantified (or classified) either on
the same location or on neighbor ones on a predefined grid. SOM are also widely
used for their more classical vector quantization property. We show in this
paper that using SOM instead of the more classical Simple Competitive Learning
(SCL) algorithm drastically increases the speed of convergence of the vector
quantization process. This fact is demonstrated through extensive simulations
on artificial and real examples, with specific SOM (fixed and decreasing
neighborhoods) and SCL algorithms.
|
math/0701144
|
Statistical tools to assess the reliability of self-organizing maps
|
math.ST cs.NE stat.TH
|
Results of neural network learning are always subject to some variability,
due to the sensitivity to initial conditions, to convergence to local minima,
and, sometimes more dramatically, to sampling variability. This paper presents
a set of tools designed to assess the reliability of the results of
Self-Organizing Maps (SOM), i.e. to test on a statistical basis the confidence
we can have on the result of a specific SOM. The tools concern the quantization
error in a SOM, and the neighborhood relations (both at the level of a specific
pair of observations and globally on the map). As a by-product, these measures
also allow to assess the adequacy of the number of units chosen in a map. The
tools may also be used to measure objectively how the SOM are less sensitive to
non-linear optimization problems (local minima, convergence, etc.) than other
neural network models.
|
math/0701145
|
Bootstrap for neural model selection
|
math.ST cs.NE stat.TH
|
Bootstrap techniques (also called resampling computation techniques) have
introduced new advances in modeling and model evaluation. Using resampling
methods to construct a series of new samples which are based on the original
data set, allows to estimate the stability of the parameters. Properties such
as convergence and asymptotic normality can be checked for any particular
observed data set. In most cases, the statistics computed on the generated data
sets give a good idea of the confidence regions of the estimates. In this
paper, we debate on the contribution of such methods for model selection, in
the case of feedforward neural networks. The method is described and compared
with the leave-one-out resampling method. The effectiveness of the bootstrap
method, versus the leave-one-out methode, is checked through a number of
examples.
|
math/0701152
|
Missing values : processing with the Kohonen algorithm
|
math.ST cs.NE stat.TH
|
The processing of data which contain missing values is a complicated and
always awkward problem, when the data come from real-world contexts. In
applications, we are very often in front of observations for which all the
values are not available, and this can occur for many reasons: typing errors,
fields left unanswered in surveys, etc. Most of the statistical software (as
SAS for example) simply suppresses incomplete observations. It has no practical
consequence when the data are very numerous. But if the number of remaining
data is too small, it can remove all significance to the results. To avoid
suppressing data in that way, it is possible to replace a missing value with
the mean value of the corresponding variable, but this approximation can be
very bad when the variable has a large variance. So it is very worthwhile
seeing that the Kohonen algorithm (as well as the Forgy algorithm) perfectly
deals with data with missing values, without having to estimate them
beforehand. We are particularly interested in the Kohonen algorithm for its
visualization properties.
|
math/0701261
|
Tracking Stopping Times Through Noisy Observations
|
math.ST cs.IT math.IT stat.TH
|
A novel quickest detection setting is proposed which is a generalization of
the well-known Bayesian change-point detection model. Suppose
\{(X_i,Y_i)\}_{i\geq 1} is a sequence of pairs of random variables, and that S
is a stopping time with respect to \{X_i\}_{i\geq 1}. The problem is to find a
stopping time T with respect to \{Y_i\}_{i\geq 1} that optimally tracks S, in
the sense that T minimizes the expected reaction delay E(T-S)^+, while keeping
the false-alarm probability P(T<S) below a given threshold \alpha \in [0,1].
This problem formulation applies in several areas, such as in communication,
detection, forecasting, and quality control.
Our results relate to the situation where the X_i's and Y_i's take values in
finite alphabets and where S is bounded by some positive integer \kappa. By
using elementary methods based on the analysis of the tree structure of
stopping times, we exhibit an algorithm that computes the optimal average
reaction delays for all \alpha \in [0,1], and constructs the associated optimal
stopping times T. Under certain conditions on \{(X_i,Y_i)\}_{i\geq 1} and S,
the algorithm running time is polynomial in \kappa.
|
math/0701419
|
Strategies for prediction under imperfect monitoring
|
math.ST cs.LG stat.TH
|
We propose simple randomized strategies for sequential prediction under
imperfect monitoring, that is, when the forecaster does not have access to the
past outcomes but rather to a feedback signal. The proposed strategies are
consistent in the sense that they achieve, asymptotically, the best possible
average reward. It was Rustichini (1999) who first proved the existence of such
consistent predictors. The forecasters presented here offer the first
constructive proof of consistency. Moreover, the proposed algorithms are
computationally efficient. We also establish upper bounds for the rates of
convergence. In the case of deterministic feedback, these rates are optimal up
to logarithmic terms.
|
math/0701791
|
Linear versus Non-linear Acquisition of Step-Functions
|
math.CA cs.CV
|
We address in this paper the following two closely related problems:
1. How to represent functions with singularities (up to a prescribed
accuracy) in a compact way?
2. How to reconstruct such functions from a small number of measurements?
The stress is on a comparison of linear and non-linear approaches. As a model
case we use piecewise-constant functions on [0,1], in particular, the Heaviside
jump function. Considered as a curve in the Hilbert space, it is completely
characterized by the fact that any two its disjoint chords are orthogonal. We
reinterpret this fact in a context of step-functions in one or two variables.
Next we study the limitations on representability and reconstruction of
piecewise-constant functions by linear and semi-linear methods. Our main tools
in this problem are Kolmogorov's n-width and entropy, as well as Temlyakov's
(N,m)-width.
On the positive side, we show that a very accurate non-linear reconstruction
is possible. It goes through a solution of certain specific non-linear systems
of algebraic equations. We discuss the form of these systems and methods of
their solution, stressing their relation to Moment Theory and Complex Analysis.
Finally, we informally discuss two problems in Computer Imaging which are
parallel to the problems 1 and 2 above: compression of still images and
video-sequences on one side, and image reconstruction from indirect measurement
(for example, in Computer Tomography), on the other.
|
math/0702116
|
A Direct Matrix Method for Computing Analytical Jacobians of Discretized
Nonlinear Integro-differential Equations
|
math.NA cs.CE
|
In this pedagogical article, we present a simple direct matrix method for
analytically computing the Jacobian of nonlinear algebraic equations that arise
from the discretization of nonlinear integro-differential equations. The method
is based on a formulation of the discretized equations in vector form using
only matrix-vector products and component-wise operations. By applying simple
matrix-based differentiation rules, the matrix form of the analytical Jacobian
can be calculated with little more difficulty than that required when computing
derivatives in single-variable calculus. After describing the direct matrix
method, we present numerical experiments demonstrating the computational
performance of the method, discuss its connection to the Newton-Kantorovich
method, and apply it to illustrative 1D and 2D example problems. MATLAB code is
provided to demonstrate the low code complexity required by the method.
|
math/0702301
|
Information-theoretic limits on sparsity recovery in the
high-dimensional and noisy setting
|
math.ST cs.IT math.IT stat.TH
|
The problem of recovering the sparsity pattern of a fixed but unknown vector
$\beta^* \in \real^p based on a set of $n$ noisy observations arises in a
variety of settings, including subset selection in regression, graphical model
selection, signal denoising, compressive sensing, and constructive
approximation. Of interest are conditions on the model dimension $p$, the
sparsity index $s$ (number of non-zero entries in $\beta^*$), and the number of
observations $n$ that are necessary and/or sufficient to ensure asymptotically
perfect recovery of the sparsity pattern. This paper focuses on the
information-theoretic limits of sparsity recovery: in particular, for a noisy
linear observation model based on measurement vectors drawn from the standard
Gaussian ensemble, we derive both a set of sufficient conditions for
asymptotically perfect recovery using the optimal decoder, as well as a set of
necessary conditions that any decoder, regardless of its computational
complexity, must satisfy for perfect recovery. This analysis of optimal
decoding limits complements our previous work (ARXIV: math.ST/0605740) on sharp
thresholds for sparsity recovery using the Lasso ($\ell_1$-constrained
quadratic programming) with Gaussian measurement ensembles.
|
math/0702804
|
The Loss Rank Principle for Model Selection
|
math.ST cs.LG stat.ME stat.ML stat.TH
|
We introduce a new principle for model selection in regression and
classification. Many regression models are controlled by some smoothness or
flexibility or complexity parameter c, e.g. the number of neighbors to be
averaged over in k nearest neighbor (kNN) regression or the polynomial degree
in regression with polynomials. Let f_D^c be the (best) regressor of complexity
c on data D. A more flexible regressor can fit more data D' well than a more
rigid one. If something (here small loss) is easy to achieve it's typically
worth less. We define the loss rank of f_D^c as the number of other
(fictitious) data D' that are fitted better by f_D'^c than D is fitted by
f_D^c. We suggest selecting the model complexity c that has minimal loss rank
(LoRP). Unlike most penalized maximum likelihood variants (AIC,BIC,MDL), LoRP
only depends on the regression function and loss function. It works without a
stochastic noise model, and is directly applicable to any non-parametric
regressor, like kNN. In this paper we formalize, discuss, and motivate LoRP,
study it for specific regression problems, in particular linear ones, and
compare it to other model selection schemes.
|
math/0702866
|
Consumer Profile Identification and Allocation
|
math.ST cs.NE stat.TH
|
We propose an easy-to-use methodology to allocate one of the groups which
have been previously built from a complete learning data base, to new
individuals. The learning data base contains continuous and categorical
variables for each individual. The groups (clusters) are built by using only
the continuous variables and described with the help of the categorical ones.
For the new individuals, only the categorical variables are available, and it
is necessary to define a model which computes the probabilities to belong to
each of the clusters, by using only the categorical variables. Then this model
provides a decision rule to assign the new individuals and gives an efficient
tool to decision-makers. This tool is shown to be very efficient for customers
allocation in consumer clusters for marketing purposes, for example.
|
math/0703019
|
Reading policies for joins: An asymptotic analysis
|
math.PR cs.DB
|
Suppose that $m_n$ observations are made from the distribution $\mathbf {R}$
and $n-m_n$ from the distribution $\mathbf {S}$. Associate with each pair, $x$
from $\mathbf {R}$ and $y$ from $\mathbf {S}$, a nonnegative score $\phi(x,y)$.
An optimal reading policy is one that yields a sequence $m_n$ that maximizes
$\mathbb{E}(M(n))$, the expected sum of the $(n-m_n)m_n$ observed scores,
uniformly in $n$. The alternating policy, which switches between the two
sources, is the optimal nonadaptive policy. In contrast, the greedy policy,
which chooses its source to maximize the expected gain on the next step, is
shown to be the optimal policy. Asymptotics are provided for the case where the
$\mathbf {R}$ and $\mathbf {S}$ distributions are discrete and $\phi(x,y)=1 or
0$ according as $x=y$ or not (i.e., the observations match). Specifically, an
invariance result is proved which guarantees that for a wide class of policies,
including the alternating and the greedy, the variable M(n) obeys the same CLT
and LIL. A more delicate analysis of the sequence $\mathbb{E}(M(n))$ and the
sample paths of M(n), for both alternating and greedy, reveals the slender
sense in which the latter policy is asymptotically superior to the former, as
well as a sense of equivalence of the two and robustness of the former.
|
math/9310227
|
A linear construction for certain Kerdock and Preparata codes
|
math.CO cs.IT math.IT
|
The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes
are shown to be linear over $\ZZ_4$, the integers $\bmod~4$. The Kerdock and
Preparata codes are duals over $\ZZ_4$, and the Nordstrom-Robinson code is
self-dual. All these codes are just extended cyclic codes over $\ZZ_4$. This
provides a simple definition for these codes and explains why their Hamming
weight distributions are dual to each other. First- and second-order
Reed-Muller codes are also linear codes over $\ZZ_4$, but Hamming codes in
general are not, nor is the Golay code.
|
math/9508219
|
New results on binary linear codes
|
math.CO cs.IT math.IT
|
This research announcement describes in very rough terms methods and a
computer language under development, which can be used to prove the
nonexistence of binary linear codes. Over a hundred new results have been
obtained by the author. For example, there is no [29,11,10] code. The proof of
this is roughly outlined.
|
math/9801152
|
On the classifiability of cellular automata
|
math.LO cs.NE
|
Based on computer simulations Wolfram presented in several papers conjectured
classifications of cellular automata into 4 types. He distinguishes the 4
classes of cellular automata by the evolution of the pattern generated by
applying a cellular automaton to a finite input. Wolfram's qualitative
classification is based on the examination of a large number of simulations. In
addition to this classification based on the rate of growth, he conjectured a
similar classification according to the eventual pattern. We consider here one
formalization of his rate of growth suggestion. After completing our major
results (based only on Wolfram's work), we investigated other contributions to
the area and we report the relation of some of them to our discoveries.
|
math/9905046
|
Duality between Multidimensional Convolutional Codes and Systems
|
math.OC cs.IT math.AC math.AG math.IT
|
Multidimensional convolutional codes generalize (one dimensional)
convolutional codes and they correspond under a natural duality to
multidimensional systems widely studied in the systems literature.
|
math/9905165
|
Perception games, the image understanding and interpretational geometry
|
math.HO cs.CV
|
The interactive game theoretical approach to the description of perception
processes is proposed. The subject is treated formally in terms of a new class
of the verbalizable interactive games which are called the perception games. An
application of the previously elaborated formalism of dialogues and
verbalizable interactive games to the visual perception allows to combine the
linguistic (such as formal grammars), psycholinguistic and (interactive) game
theoretical methods for analysis of the image understanding by a human that may
be also useful for the elaboration of computer vision systems. By the way the
interactive game theoretical aspects of interpretational geometries are
clarified.
|
math/9909163
|
Coding Theory and Uniform Distributions
|
math.CO cs.IT math.IT math.NT
|
In the present paper we introduce and study finite point subsets of a special
kind, called optimum distributions, in the n-dimensional unit cube. Such
distributions are closely related with known (delta,s,n)-nets of low
discrepancy. It turns out that optimum distributions have a rich combinatorial
structure. Namely, we show that optimum distributions can be characterized
completely as maximum distance separable codes with respect to a non-Hamming
metric. Weight spectra of such codes can be evaluated precisely. We also
consider linear codes and distributions and study their general properties
including the duality with respect to a suitable inner product. The
corresponding generalized MacWilliams identities for weight enumerators are
briefly discussed. Broad classes of linear maximum distance separable codes and
linear optimum distributions are explicitely constructed in the paper by the
Hermite interpolations over finite fields.
|
math/9910062
|
Efficient sphere-covering and converse measure concentration via
generalized coding theorems
|
math.PR cs.IT math.FA math.IT
|
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.
|
math/9910149
|
Rational points, genus and asymptotic behaviour in reduced algebraic
curves over finite fields
|
math.AG cs.IT math.IT math.NT
|
The number A(q) shows the asymptotic behaviour of the quotient of the number
of rational points over the genus of non-singular absolutely irreducible curves
over a finite field Fq. Research on bounds for A(q) is closely connected with
the so-called asymptotic main problem in Coding Theory. In this paper, we study
some generalizations of this number for non-irreducible curves, their
connection with A(q) and its application in Coding Theory.
|
math/9910151
|
Decoding Algebraic Geometry codes by a key equation
|
math.AG cs.IT math.IT math.NA
|
A new effective decoding algorithm is presented for arbitrary
algebraic-geometric codes on the basis of solving a generalized key equation
with the majority coset scheme of Duursma. It is an improvement of Ehrhard's
algorithm, since the method corrects up to the half of the Goppa distance with
complexity order O(n**2.81), and with no further assumption on the degree of
the divisor G.
|
math/9910155
|
Computing Weierstrass semigroups and the Feng-Rao distance from singular
plane models
|
math.AG cs.IT math.IT math.NA
|
We present an algorithm to compute the Weierstrass semigroup at a point P
together with functions for each value in the semigroup, provided P is the only
branch at infinity of a singular plane model for the curve. As a byproduct, the
method also provides us with a basis for the spaces L(mP) and the computation
of the Feng-Rao distance for the corresponding array of geometric Goppa codes.
A general computation of the Feng-Rao distance is also obtained. Everything can
be applied to the decoding problem by using the majority scheme of Feng and
Rao.
|
math/9910175
|
Polynomial method in coding and information theory
|
math.CO cs.IT math.IT
|
Polynomial, or Delsarte's, method in coding theory accounts for a variety of
structural results on, and bounds on the size of, extremal configurations
(codes and designs) in various metric spaces. In recent works of the authors
the applicability of the method was extended to cover a wider range of problems
in coding and information theory. In this paper we present a general framework
for the method which includes previous results as particular cases. We explain
how this generalization leads to new asymptotic bounds on the performance of
codes in binary-input memoryless channels and the Gaussian channel, which
improve the results of Shannon et al. of 1959-67, and to a number of other
results in combinatorial coding theory.
|
math/9911025
|
On the parameters of Algebraic Geometry codes related to Arf semigroups
|
math.NT cs.IT math.AG math.IT
|
In this paper we compute the order (or Feng-Rao) bound on the minimum
distance of one-point algebraic geometry codes, when the Weierstrass semigroup
at the point Q is an Arf semigroup. The results developed to that purpose also
provide the dimension of the improved geometric Goppa codes related to those.
|
nlin/0001057
|
Numerical Replication of Computer Simulations: Some Pitfalls and How To
Avoid Them
|
nlin.AO cs.NE
|
A computer simulation, such as a genetic algorithm, that uses IEEE standard
floating-point arithmetic may not produce exactly the same results in two
different runs, even if it is rerun on the same computer with the same input
and random number seeds. Researchers should not simply assume that the results
from one run replicate those from another but should verify this by actually
comparing the data. However, researchers who are aware of this pitfall can
reliably replicate simulations, in practice. This paper discusses the problem
and suggests solutions.
|
nlin/0002040
|
The dynamics of iterated transportation simulations
|
nlin.AO cs.CE
|
Iterating between a router and a traffic micro-simulation is an increasibly
accepted method for doing traffic assignment. This paper, after pointing out
that the analytical theory of simulation-based assignment to-date is
insufficient for some practical cases, presents results of simulation studies
from a real world study. Specifically, we look into the issues of uniqueness,
variability, and robustness and validation. Regarding uniqueness, despite some
cautionary notes from a theoretical point of view, we find no indication of
``meta-stable'' states for the iterations. Variability however is considerable.
By variability we mean the variation of the simulation of a given plan set by
just changing the random seed. We show then results from three different
micro-simulations under the same iteration scenario in order to test for the
robustness of the results under different implementations. We find the results
encouraging, also when comparing to reality and with a traditional assignment
result.
Keywords: dynamic traffic assignment (DTA); traffic micro-simulation;
TRANSIMS; large-scale simulations; urban planning
|
nlin/0006025
|
Information Bottlenecks, Causal States, and Statistical Relevance Bases:
How to Represent Relevant Information in Memoryless Transduction
|
nlin.AO cond-mat.dis-nn cs.LG physics.data-an
|
Discovering relevant, but possibly hidden, variables is a key step in
constructing useful and predictive theories about the natural world. This brief
note explains the connections between three approaches to this problem: the
recently introduced information-bottleneck method, the computational mechanics
approach to inferring optimal models, and Salmon's statistical relevance basis.
|
nlin/0109019
|
Olfactory search at high Reynolds number
|
nlin.CD cs.RO nlin.AO physics.bio-ph
|
Locating the source of odor in a turbulent environment - a common behavior
for living organisms - is non-trivial because of the random nature of mixing.
Here we analyze the statistical physics aspects of the problem and propose an
efficient strategy for olfactory search which can work in turbulent plumes. The
algorithm combines the maximum likelihood inference of the source position with
an active search. Our approach provides the theoretical basis for the design of
olfactory robots and the quantitative tools for the analysis of the observed
olfactory search behavior of living creatures (e.g. odor modulated optomotor
anemotaxis of moth)
|
nlin/0202038
|
On model selection and the disability of neural networks to decompose
tasks
|
nlin.AO cond-mat.dis-nn cs.NE
|
A neural network with fixed topology can be regarded as a parametrization of
functions, which decides on the correlations between functional variations when
parameters are adapted. We propose an analysis, based on a differential
geometry point of view, that allows to calculate these correlations. In
practise, this describes how one response is unlearned while another is
trained. Concerning conventional feed-forward neural networks we find that they
generically introduce strong correlations, are predisposed to forgetting, and
inappropriate for task decomposition. Perspectives to solve these problems are
discussed.
|
nlin/0202039
|
A neural model for multi-expert architectures
|
nlin.AO cond-mat.dis-nn cs.NE
|
We present a generalization of conventional artificial neural networks that
allows for a functional equivalence to multi-expert systems. The new model
provides an architectural freedom going beyond existing multi-expert models and
an integrative formalism to compare and combine various techniques of learning.
(We consider gradient, EM, reinforcement, and unsupervised learning.) Its
uniform representation aims at a simple genetic encoding and evolutionary
structure optimization of multi-expert systems. This paper contains a detailed
description of the model and learning rules, empirically validates its
functionality, and discusses future perspectives.
|
nlin/0204038
|
Neutrality: A Necessity for Self-Adaptation
|
nlin.AO cs.NE q-bio
|
Self-adaptation is used in all main paradigms of evolutionary computation to
increase efficiency. We claim that the basis of self-adaptation is the use of
neutrality. In the absence of external control neutrality allows a variation of
the search distribution without the risk of fitness loss.
|
nlin/0210041
|
Simple Model for the Dynamics of Correlations in the Evolution of
Economic Entities Under Varying Economic Conditions
|
nlin.AO cond-mat.stat-mech cs.CE physics.soc-ph
|
From some observations on economic behaviors, in particular changing economic
conditions with time and space, we develop a very simple model for the
evolution of economic entities within a geographical type of framework. We
raise a few questions and attempt to investigate whether some of them can be
tackled by our model. Several cases of interest are reported. It is found that
the model even in its simple forms can lead to a large variety of situations,
including: delocalization and cycles, but also pre-chaotic behavior.
|
nlin/0211010
|
Evolution and anti-evolution in a minimal stock market model
|
nlin.AO cond-mat.stat-mech cs.MA q-fin.TR
|
We present a novel microscopic stock market model consisting of a large
number of random agents modeling traders in a market. Each agent is
characterized by a set of parameters that serve to make iterated predictions of
two successive returns. The future price is determined according to the offer
and the demand of all agents. The system evolves by redistributing the capital
among the agents in each trading cycle. Without noise the dynamics of this
system is nearly regular and thereby fails to reproduce the stochastic return
fluctuations observed in real markets. However, when in each cycle a small
amount of noise is introduced we find the typical features of real financial
time series like fat-tails of the return distribution and large temporal
correlations in the volatility without significant correlations in the price
returns. Introducing the noise by an evolutionary process leads to different
scalings of the return distributions that depend on the definition of fitness.
Because our realistic model has only very few parameters, and the results
appear to be robust with respect to the noise level and the number of agents we
expect that our framework may serve as new paradigm for modeling self generated
return fluctuations in markets.
|
nlin/0211013
|
A Spin Glass Model of Human Logic Systems
|
nlin.AO cond-mat.dis-nn cs.MA
|
In this paper, we discuss different models for human logic systems and
describe a game with nature. G\"odel`s incompleteness theorem is taken into
account to construct a model of logical networks based on axioms obtained by
symmetry breaking. These classical logic networks are then coupled using rules
that depend on whether two networks contain axioms or anti-axioms. The social
lattice of axiom based logic networks is then placed with the environment
network in a game including entropy as a cost factor. The classical logical
networks are then replaced with ``preference axioms'' to the role of fuzzy
logic.
|
nlin/0211024
|
Exploring the cooperative regimes in a model of agents without memory or
"tags": indirect reciprocity vs. selfish incentives
|
nlin.AO cond-mat cs.CE hep-lat nlin.CG physics.soc-ph
|
The self-organization in cooperative regimes in a simple mean-field version
of a model based on "selfish" agents which play the Prisoner's Dilemma (PD)
game is studied. The agents have no memory and use strategies not based on
direct reciprocity nor 'tags'. Two variables are assigned to each agent $i$ at
time $t$, measuring its capital $C(i;t)$ and its probability of cooperation
$p(i;t)$. At each time step $t$ a pair of agents interact by playing the PD
game. These 2 agents update their probability of cooperation $p(i)$ as follows:
they compare the profits they made in this interaction $\delta C(i;t)$ with an
estimator $\epsilon(i;t)$ and, if $\delta C(i;t) \ge \epsilon(i;t)$, agent $i$
increases its $p(i;t)$ while if $\delta C(i;t) < \epsilon(i;t)$ the agent
decreases $p(i;t)$. The 4!=24 different cases produced by permuting the four
Prisoner's Dilemma canonical payoffs 3, 0, 1, and 5 -
corresponding,respectively, to $R$ (reward), $S$ (sucker's payoff), $T$
(temptation to defect) and $P$ (punishment) - are analyzed. It turns out that
for all these 24 possibilities, after a transient,the system self-organizes
into a stationary state with average equilibrium probability of cooperation
$\bar{p}_\infty$ = constant $ > 0$.Depending on the payoff matrix, there are
different equilibrium states characterized by their average probability of
cooperation and average equilibrium per-capita-income
($\bar{p}_\infty,\bar{\delta C}_\infty$).
|
nlin/0212030
|
The structure of evolutionary exploration: On crossover, buildings
blocks and Estimation-Of-Distribution Algorithms
|
nlin.AO cs.NE q-bio
|
The notion of building blocks can be related to the structure of the
offspring probability distribution: loci of which variability is strongly
correlated constitute a building block. We call this correlated exploration.
With this background we analyze the structure of the offspring probability
distribution, or exploration distribution, for a GA with mutation only, a
crossover GA, and an Estimation-Of-Distribution Algorithm (EDA). The results
allow a precise characterization of the structure of the crossover exploration
distribution. Essentially, the crossover operator destroys mutual information
between loci by transforming it into entropy; it does the inverse of correlated
exploration. In contrast, the objective of EDAs is to model the mutual
information between loci in the fitness distribution and thereby they induce
correlated exploration.
|
nlin/0304006
|
Determining possible avenues of approach using ANTS
|
nlin.AO cs.AI
|
Threat assessment is an important part of level 3 data fusion. Here we study
a subproblem of this, worst-case risk assessment. Inspired by agent-based
models used for simulation of trail formation for urban planning, we use ant
colony optimization (ANTS) to determine possible avenues of approach for the
enemy, given a situation picture.
One way of determining such avenues would be to calculate the ``potential
field'' caused by placing sources at possible goals for the enemy. This
requires postulating a functional form for the potential, and also takes long
time. Here we instead seek a method for quickly obtaining an effective
potential. ANTS, which has previously been used to obtain approximate solutions
to various optimization problems, is well suited for this. The output of our
method describes possible avenues of approach for the enemy, i.e, areas where
we should be prepared for attack. (The algorithm can also be run ``reversed''
to instead get areas of opportunity for our forces to exploit.)
Using real geographical data, we found that our method gives a fast and
reliable way of determining such avenues. Our method can be used in a
computer-based command and control system to replace the first step of human
intelligence analysis.
|
nlin/0306055
|
A Model for Prejudiced Learning in Noisy Environments
|
nlin.AO cs.LG
|
Based on the heuristics that maintaining presumptions can be beneficial in
uncertain environments, we propose a set of basic axioms for learning systems
to incorporate the concept of prejudice. The simplest, memoryless model of a
deterministic learning rule obeying the axioms is constructed, and shown to be
equivalent to the logistic map. The system's performance is analysed in an
environment in which it is subject to external randomness, weighing learning
defectiveness against stability gained. The corresponding random dynamical
system with inhomogeneous, additive noise is studied, and shown to exhibit the
phenomena of noise induced stability and stochastic bifurcations. The overall
results allow for the interpretation that prejudice in uncertain environments
entails a considerable portion of stubbornness as a secondary phenomenon.
|
nlin/0309039
|
Self-organizing Traffic Control: First Results
|
nlin.AO cs.MA
|
We developed a virtual laboratory for traffic control where agents use
different strategies in order to self-organize on the road. We present our
first results where we compare the performance and behaviour promoted by
environmental constrains and five different simple strategies: three inspired
in flocking behaviour, one selfish, and one inspired in the minority game.
Experiments are presented for comparing the strategies. Different issues are
discussed, such as the important role of environmental constrains and the
emergence of traffic lanes.
|
nlin/0312056
|
Shannon information, LMC complexity and Renyi entropies: a
straightforward approach
|
nlin.AO cond-mat.stat-mech cs.IT math.IT nlin.CD physics.comp-ph q-bio.QM
|
The LMC complexity, an indicator of complexity based on a probabilistic
description, is revisited. A straightforward approach allows us to establish
the time evolution of this indicator in a near-equilibrium situation and gives
us a new insight for interpreting the LMC complexity for a general non
equilibrium system. Its relationship with the Renyi entropies is also
explained. One of the advantages of this indicator is that its calculation does
not require a considerable computational effort in many cases of physical and
biological interest.
|
nlin/0402046
|
Spontaneous Emergence of Complex Optimal Networks through Evolutionary
Adaptation
|
nlin.AO cond-mat.stat-mech cs.MA nlin.CG q-bio.QM
|
An important feature of many complex systems, both natural and artificial, is
the structure and organization of their interaction networks with interesting
properties. Here we present a theory of self-organization by evolutionary
adaptation in which we show how the structure and organization of a network is
related to the survival, or in general the performance, objectives of the
system. We propose that a complex system optimizes its network structure in
order to maximize its overall survival fitness which is composed of short-term
and long-term survival components. These in turn depend on three critical
measures of the network, namely, efficiency, robustness and cost, and the
environmental selection pressure. Using a graph theoretical case study, we show
that when efficiency is paramount the "Star" topology emerges and when
robustness is important the "Circle" topology is found. When efficiency and
robustness requirements are both important to varying degrees, other classes of
networks such as the "Hub" emerge. Our assumptions and results are consistent
with observations across a wide variety of applications.
|
nlin/0404004
|
Protocol Requirements for Self-organizing Artifacts: Towards an Ambient
Intelligence
|
nlin.AO cs.AI
|
We discuss which properties common-use artifacts should have to collaborate
without human intervention. We conceive how devices, such as mobile phones,
PDAs, and home appliances, could be seamlessly integrated to provide an
"ambient intelligence" that responds to the user's desires without requiring
explicit programming or commands. While the hardware and software technology to
build such systems already exists, as yet there is no standard protocol that
can learn new meanings. We propose the first steps in the development of such a
protocol, which would need to be adaptive, extensible, and open to the
community, while promoting self-organization. We argue that devices,
interacting through "game-like" moves, can learn to agree about how to
communicate, with whom to cooperate, and how to delegate and coordinate
specialized tasks. Thus, they may evolve a distributed cognition or collective
intelligence capable of tackling complex tasks.
|
nlin/0404032
|
Metrics for more than two points at once
|
nlin.AO cond-mat.other cs.LG math.GM
|
The conventional definition of a topological metric over a space specifies
properties that must be obeyed by any measure of "how separated" two points in
that space are. Here it is shown how to extend that definition, and in
particular the triangle inequality, to concern arbitrary numbers of points.
Such a measure of how separated the points within a collection are can be
bootstrapped, to measure "how separated" from each other are two (or more)
collections. The measure presented here also allows fractional membership of an
element in a collection. This means it directly concerns measures of ``how
spread out" a probability distribution over a space is. When such a measure is
bootstrapped to compare two collections, it allows us to measure how separated
two probability distributions are, or more generally, how separated a
distribution of distributions is.
|
nlin/0407032
|
Application of Artificial Neural Network in Jitter Analysis of
Dispersion-Managed Communication System
|
nlin.PS cs.AI cs.NA
|
Artificial Neural Network (ANN) is used as numerical methode in solving
modified Nonlinear Schroedinger (NLS) equation with Dispersion Managed System
(DMS) for jitter analysis. We take the optical axis z and the time t as input,
and then some relevant values such as the change of position and the center
frequency of the pulse, and further the mean square time of incoming pulse
which are needed for jitter analysis. It shows that ANN yields numerical
solutions which are adaptive with respect to the numerical errors and also
verifies the previous numerical results using conventional numerical method.
Our result indicates that DMS can minimize the timing jitter induced by some
amplifiers.
|
nlin/0408007
|
Entropy Maximization as a Holistic Design Principle for Complex Optimal
Networks and the Emergence of Power Laws
|
nlin.AO cond-mat.stat-mech cs.IT math.IT q-bio.QM
|
We present a general holistic theory for the organization of complex
networks, both human-engineered and naturally-evolved. Introducing concepts of
value of interactions and satisfaction as generic network performance measures,
we show that the underlying organizing principle is to meet an overall
performance target for wide-ranging operating or environmental conditions. This
design or survival requirement of reliable performance under uncertainty leads,
via the maximum entropy principle, to the emergence of a power law vertex
degree distribution. The theory also predicts exponential or Poisson degree
distributions depending on network redundancy, thus explaining all three
regimes as different manifestations of a common underlying phenomenon within a
unified theoretical framework.
|
nlin/0408039
|
Stability and Diversity in Collective Adaptation
|
nlin.AO cs.LG math.DS nlin.CD stat.ML
|
We derive a class of macroscopic differential equations that describe
collective adaptation, starting from a discrete-time stochastic microscopic
model. The behavior of each agent is a dynamic balance between adaptation that
locally achieves the best action and memory loss that leads to randomized
behavior. We show that, although individual agents interact with their
environment and other agents in a purely self-interested way, macroscopic
behavior can be interpreted as game dynamics. Application to several familiar,
explicit game interactions shows that the adaptation dynamics exhibits a
diversity of collective behaviors. The simplicity of the assumptions underlying
the macroscopic equations suggests that these behaviors should be expected
broadly in collective adaptation. We also analyze the adaptation dynamics from
an information-theoretic viewpoint and discuss self-organization induced by
information flux between agents, giving a novel view of collective adaptation.
|
nlin/0408040
|
Notes on information geometry and evolutionary processes
|
nlin.AO cs.NE
|
In order to analyze and extract different structural properties of
distributions, one can introduce different coordinate systems over the manifold
of distributions. In Evolutionary Computation, the Walsh bases and the Building
Block Bases are often used to describe populations, which simplifies the
analysis of evolutionary operators applying on populations. Quite independent
from these approaches, information geometry has been developed as a geometric
way to analyze different order dependencies between random variables (e.g.,
neural activations or genes).
In these notes I briefly review the essentials of various coordinate bases
and of information geometry. The goal is to give an overview and make the
approaches comparable. Besides introducing meaningful coordinate bases,
information geometry also offers an explicit way to distinguish different order
interactions and it offers a geometric view on the manifold and thereby also on
operators that apply on the manifold. For instance, uniform crossover can be
interpreted as an orthogonal projection of a population along an m-geodesic,
monotonously reducing the theta-coordinates that describe interactions between
genes.
|
nlin/0409013
|
Epistemic communities: description and hierarchic categorization
|
nlin.AO cs.IR
|
Social scientists have shown an increasing interest in understanding the
structure of knowledge communities, and particularly the organization of
"epistemic communities", that is groups of agents sharing common knowledge
concerns. However, most existing approaches are based only on either social
relationships or semantic similarity, while there has been roughly no attempt
to link social and semantic aspects. In this paper, we introduce a formal
framework addressing this issue and propose a method based on Galois lattices
(or concept lattices) for categorizing epistemic communities in an automated
and hierarchically structured fashion. Suggesting that our process allows us to
rebuild a whole community structure and taxonomy, and notably fields and
subfields gathering a certain proportion of agents, we eventually apply it to
empirical data to exhibit these alleged structural properties, and successfully
compare our results with categories spontaneously given by domain experts.
|
nlin/0411063
|
Detecting synchronization in spatially extended discrete systems by
complexity measurements
|
nlin.CG cond-mat.dis-nn cs.MA math.DS nlin.PS q-bio.QM
|
The synchronization of two stochastically coupled one-dimensional cellular
automata (CA) is analyzed. It is shown that the transition to synchronization
is characterized by a dramatic increase of the statistical complexity of the
patterns generated by the difference automaton. This singular behavior is
verified to be present in several CA rules displaying complex behavior.
|
nlin/0411066
|
Self-Organizing Traffic Lights
|
nlin.AO cond-mat.stat-mech cs.AI cs.MA
|
Steering traffic in cities is a very complex task, since improving efficiency
involves the coordination of many actors. Traditional approaches attempt to
optimize traffic lights for a particular density and configuration of traffic.
The disadvantage of this lies in the fact that traffic densities and
configurations change constantly. Traffic seems to be an adaptation problem
rather than an optimization problem. We propose a simple and feasible
alternative, in which traffic lights self-organize to improve traffic flow. We
use a multi-agent simulation to study three self-organizing methods, which are
able to outperform traditional rigid and adaptive methods. Using simple rules
and no direct communication, traffic lights are able to self-organize and adapt
to changing traffic conditions, reducing waiting times, number of stopped cars,
and increasing average speeds.
|
nlin/0505043
|
A network analysis of committees in the United States House of
Representatives
|
nlin.AO cs.MA math.ST physics.data-an physics.soc-ph stat.TH
|
Network theory provides a powerful tool for the representation and analysis
of complex systems of interacting agents. Here we investigate the United States
House of Representatives network of committees and subcommittees, with
committees connected according to ``interlocks'' or common membership. Analysis
of this network reveals clearly the strong links between different committees,
as well as the intrinsic hierarchical structure within the House as a whole. We
show that network theory, combined with the analysis of roll call votes using
singular value decomposition, successfully uncovers political and
organizational correlations between committees in the House without the need to
incorporate other political information.
|
nlin/0506061
|
Transmitting a signal by amplitude modulation in a chaotic network
|
nlin.CD cond-mat.stat-mech cs.NE
|
We discuss the ability of a network with non linear relays and chaotic
dynamics to transmit signals, on the basis of a linear response theory
developed by Ruelle \cite{Ruelle} for dissipative systems. We show in
particular how the dynamics interfere with the graph topology to produce an
effective transmission network, whose topology depends on the signal, and
cannot be directly read on the ``wired'' network. This leads one to reconsider
notions such as ``hubs''. Then, we show examples where, with a suitable choice
of the carrier frequency (resonance), one can transmit a signal from a node to
another one by amplitude modulation, \textit{in spite of chaos}. Also, we give
an example where a signal, transmitted to any node via different paths, can
only be recovered by a couple of \textit{specific} nodes. This opens the
possibility for encoding data in a way such that the recovery of the signal
requires the knowledge of the carrier frequency \textit{and} can be performed
only at some specific node.
|
nlin/0508006
|
Metamimetic Games : Modeling Metadynamics in Social Cognition
|
nlin.AO cs.MA nlin.CG
|
Imitation is fundamental in the understanding of social system dynamics. But
the diversity of imitation rules employed by modelers proves that the modeling
of mimetic processes cannot avoid the traditional problem of endogenization of
all the choices, including the one of the mimetic rules. Starting from the
remark that human reflexive capacities are the ground for a new class of
mimetic rules, I propose a formal framework, metamimetic games, that enable to
endogenize the distribution of imitation rules while being human specific. The
corresponding concepts of equilibrium - counterfactually stable state - and
attractor are introduced. Finally, I give an interpretation of social
differentiation in terms of cultural co-evolution among a set of possible
motivations, which departs from the traditional view of optimization indexed to
criteria that exist prior to the activity of agents.
|
nlin/0509007
|
Lattices for Dynamic, Hierarchic & Overlapping Categorization: the Case
of Epistemic Communities
|
nlin.AO cs.AI cs.DL cs.IR
|
We present a method for hierarchic categorization and taxonomy evolution
description. We focus on the structure of epistemic communities (ECs), or
groups of agents sharing common knowledge concerns. Introducing a formal
framework based on Galois lattices, we categorize ECs in an automated and
hierarchically structured way and propose criteria for selecting the most
relevant epistemic communities - for instance, ECs gathering a certain
proportion of agents and thus prototypical of major fields. This process
produces a manageable, insightful taxonomy of the community. Then, the
longitudinal study of these static pictures makes possible an historical
description. In particular, we capture stylized facts such as field progress,
decline, specialization, interaction (merging or splitting), and paradigm
emergence. The detection of such patterns in social networks could fruitfully
be applied to other contexts.
|
nlin/0511015
|
Combinatorial Approach to Object Analysis
|
nlin.AO cs.LG
|
We present a perceptional mathematical model for image and signal analysis. A
resemblance measure is defined, and submitted to an innovating combinatorial
optimization algorithm. Numerical Simulations are also presented
|
nlin/0512048
|
Modeling Endogenous Social Networks: the Example of Emergence and
Stability of Cooperation without Refusal
|
nlin.AO cond-mat.other cs.GT cs.MA cs.OH q-bio.OT q-bio.PE
|
Aggregated phenomena in social sciences and economics are highly dependent on
the way individuals interact. To help understanding the interplay between
socio-economic activities and underlying social networks, this paper studies a
sequential prisoner's dilemma with binary choice. It proposes an analytical and
computational insight about the role of endogenous networks in emergence and
sustainability of cooperation and exhibits an alternative to the choice and
refusal mechanism that is often proposed to explain cooperation. The study
focuses on heterogeneous equilibriums and emergence of cooperation from an
all-defector state that are the two stylized facts that this model successfully
reconstructs.
|
nlin/0605029
|
Three Logistic Models for the Ecological and Economic Interactions:
Symbiosis, Predator-Prey and Competition
|
nlin.AO cs.MA math.DS
|
If one isolated species (corporation) is supposed to evolve following the
logistic mapping, then we are tempted to think that the dynamics of two species
(corporations) can be expressed by a coupled system of two discrete logistic
equations. As three basic relationships between two species are present in
Nature, namely symbiosis, predator-prey and competition, three different models
are obtained. Each model is a cubic two-dimensional discrete logistic-type
equation with its own dynamical properties: stationary regime, periodicity,
quasi-periodicity and chaos. We also propose that these models could be useful
for thinking in the different interactions happening in the economic world, as
for instance for the competition and the collaboration between corporations.
Furthermore, these models could be considered as the basic ingredients to
construct more complex interactions in the ecological and economic networks.
|
nlin/0609033
|
Fame Emerges as a Result of Small Memory
|
nlin.AO cs.CY cs.MA physics.soc-ph
|
A dynamic memory model is proposed in which an agent ``learns'' a new agent
by means of recommendation. The agents can also ``remember'' and ``forget''.
The memory size is decreased while the population size is kept constant.
``Fame'' emerged as a few agents become very well known in expense of the
majority being completely forgotten. The minimum and the maximum of fame change
linearly with the relative memory size. The network properties of the
who-knows-who graph, which represents the state of the system, are
investigated.
|
nlin/0609038
|
From Neuron to Neural Networks dynamics
|
nlin.AO cond-mat.dis-nn cs.NE
|
This paper presents an overview of some techniques and concepts coming from
dynamical system theory and used for the analysis of dynamical neural networks
models. In a first section, we describe the dynamics of the neuron, starting
from the Hodgkin-Huxley description, which is somehow the canonical description
for the ``biological neuron''. We discuss some models reducing the
Hodgkin-Huxley model to a two dimensional dynamical system, keeping one of the
main feature of the neuron: its excitability. We present then examples of phase
diagram and bifurcation analysis for the Hodgin-Huxley equations. Finally, we
end this section by a dynamical system analysis for the nervous flux
propagation along the axon. We then consider neuron couplings, with a brief
description of synapses, synaptic plasticiy and learning, in a second section.
We also briefly discuss the delicate issue of causal action from one neuron to
another when complex feedback effects and non linear dynamics are involved. The
third section presents the limit of weak coupling and the use of normal forms
technics to handle this situation. We consider then several examples of
recurrent models with different type of synaptic interactions (symmetric,
cooperative, random). We introduce various techniques coming from statistical
physics and dynamical systems theory. A last section is devoted to a detailed
example of recurrent model where we go in deep in the analysis of the dynamics
and discuss the effect of learning on the neuron dynamics. We also present
recent methods allowing the analysis of the non linear effects of the neural
dynamics on signal propagation and causal action. An appendix, presenting the
main notions of dynamical systems theory useful for the comprehension of the
chapter, has been added for the convenience of the reader.
|
nlin/0610040
|
Self-organizing traffic lights: A realistic simulation
|
nlin.AO cond-mat.stat-mech cs.AI physics.comp-ph physics.soc-ph
|
We have previously shown in an abstract simulation (Gershenson, 2005) that
self-organizing traffic lights can improve greatly traffic flow for any
density. In this paper, we extend these results to a realistic setting,
implementing self-organizing traffic lights in an advanced traffic simulator
using real data from a Brussels avenue. On average, for different traffic
densities, travel waiting times are reduced by 50% compared to the current
green wave method.
|
nlin/0611044
|
Why the Maxwellian Distribution is the Attractive Fixed Point of the
Boltzmann Equation
|
nlin.CD cond-mat.stat-mech cs.MA math.ST physics.class-ph stat.TH
|
The origin of the Boltzmann factor is revisited. An alternative derivation
from the microcanonical picture is given. The Maxwellian distribution in a
mono-dimensional ideal gas is obtained by following this insight. Other
possible applications, as for instance the obtaining of the wealth distribution
in the human society, are suggested in the remarks.
|
nlin/0611054
|
A Model of a Trust-based Recommendation System on a Social Network
|
nlin.AO cs.IR physics.soc-ph
|
In this paper, we present a model of a trust-based recommendation system on a
social network. The idea of the model is that agents use their social network
to reach information and their trust relationships to filter it. We investigate
how the dynamics of trust among agents affect the performance of the system by
comparing it to a frequency-based recommendation system. Furthermore, we
identify the impact of network density, preference heterogeneity among agents,
and knowledge sparseness to be crucial factors for the performance of the
system. The system self-organises in a state with performance near to the
optimum; the performance on the global level is an emergent property of the
system, achieved without explicit coordination from the local interactions of
agents.
|
nlin/0702001
|
Bistability: a common feature in some "aggregates" of logistic maps
|
nlin.AO cs.NE
|
As it was argued by Anderson [Science 177, 393 (1972)], the "reductionist"
hypothesis does not by any means imply a "constructionist" one. Hence, in
general, the behavior of large and complex aggregates of elementary components
can not be understood nor extrapolated from the properties of a few components.
Following this insight, we have simulated different "aggregates" of logistic
maps according to a particular coupling scheme. All these aggregates show a
similar pattern of dynamical properties, concretely a bistable behavior, that
is also found in a network of many units of the same type, independently of the
number of components and of the interconnection topology. A qualitative
relationship with brain-like systems is suggested.
|
nlin/0703036
|
Statistical User Model for the Internet Access
|
nlin.AO cond-mat.stat-mech cs.MA cs.NI
|
A new statistical based model approach to characterize a user's behavior in
an Internet access link is presented. The real patterns of Internet traffic in
a heterogeneous Campus Network are studied. We find three clearly different
patterns of individual user's behavior, study their common features and group
particular users behaving alike in three clusters. This allows us to build a
probabilistic mixture model, that can explain the expected global behavior for
the three different types of users. We discuss the implications of this
emergent phenomenology in the field of multi-agent complex systems.
|
nlin/0703050
|
Competition of Self-Organized Rotating Spiral Autowaves in a
Nonequilibrium Dissipative System of Three-Level Phaser
|
nlin.CG cs.NE nlin.AO
|
We present results of cellular automata based investigations of rotating
spiral autowaves in a nonequilibrium excitable medium which models three-level
paramagnetic microwave phonon laser (phaser). The computational model is
described in arXiv:cond-mat/0410460v2 and arXiv:cond-mat/0602345v1 . We have
observed several new scenarios of self-organization, competition and dynamical
stabilization of rotating spiral autowaves under conditions of cross-relaxation
between three-level active centers. In particular, phenomena of inversion of
topological charge, as well as processes of regeneration and replication of
rotating spiral autowaves in various excitable media were revealed and
visualized for mesoscopic-scale areas of phaser-type active systems, which
model real phaser devices.
|
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