Convergence in distribution of the multi-dimensional Kohonen algorithm

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

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MULTIPLICATIVE STOCHASTIC PERTURBATIONS OF ONE-DIMENSIONAL MAPS

Stochastics and Dynamics ◽

10.1142/s0219493712500207 ◽

2013 ◽

Vol 13 (02) ◽

pp. 1250020 ◽

Cited By ~ 1

Author(s):

YUKIKO IWATA

Keyword(s):

Markov Process ◽

Probability Measure ◽

Noise Level ◽

Random Perturbation ◽

Invariant Probability Measure ◽

Random Perturbations ◽

Stochastic Perturbations ◽

One Dimensional ◽

One Dimensional Maps

We consider random perturbations of some one-dimensional map S : [0, 1] → [0, 1] such that [Formula: see text] parametrized by 0 < ε < 1, where {Cn} is an i.i.d. sequence. We prove that this random perturbation is small with respect to the noise level 0 < ε < 1 and give a class of one-dimensional maps for which there always exists a smooth invariant probability measure for the Markov process {Xn}n≥0.

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Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

Advances in Applied Probability ◽

10.2307/1427581 ◽

1994 ◽

Vol 26 (1) ◽

pp. 80-103 ◽

Cited By ~ 20

Author(s):

Catherine Bouton ◽

Gilles Pagès

Keyword(s):

Markov Chain ◽

Probability Measure ◽

Asymptotic Behaviour ◽

Absolute Continuity ◽

Invariant Probability Measure ◽

Convergence In Distribution ◽

One Dimensional ◽

Open Set ◽

Recurrent Markov Chain ◽

The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

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Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

Advances in Applied Probability ◽

10.1017/s0001867800026021 ◽

1994 ◽

Vol 26 (01) ◽

pp. 80-103 ◽

Cited By ~ 3

Author(s):

Catherine Bouton ◽

Gilles Pagès

Keyword(s):

Markov Chain ◽

Probability Measure ◽

Asymptotic Behaviour ◽

Absolute Continuity ◽

Invariant Probability Measure ◽

Convergence In Distribution ◽

One Dimensional ◽

Open Set ◽

Recurrent Markov Chain ◽

The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

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The structure of tame minimal dynamical systems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385707000296 ◽

2007 ◽

Vol 27 (6) ◽

pp. 1819-1837 ◽

Cited By ~ 24

Author(s):

ELI GLASNER

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Probability Measure ◽

Topological Space ◽

Haar Measure ◽

Invariant Probability Measure ◽

Structure Theory ◽

Convergence Of Sequences ◽

Minimal Dynamical Systems ◽

Minimal Dynamical System

AbstractA dynamical version of the Bourgain–Fremlin–Talagrand dichotomy shows that the enveloping semigroup of a dynamical system is either very large and contains a topological copy of $\beta \mathbb {N}$, or it is a ‘tame’ topological space whose topology is determined by the convergence of sequences. In the latter case, the dynamical system is said to be tame. We use the structure theory of minimal dynamical systems to show that, when the acting group is Abelian, a tame metric minimal dynamical system (i) is almost automorphic (i.e. it is an almost one-to-one extension of an equicontinuous system), and (ii) admits a unique invariant probability measure such that the corresponding measure-preserving system is measure-theoretically isomorphic to the Haar measure system on the maximal equicontinuous factor.

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Ergodic properties and ergodic decompositions of continuous-time Markov processes

Journal of Applied Probability ◽

10.1239/jap/1158784945 ◽

2006 ◽

Vol 43 (3) ◽

pp. 767-781 ◽

Cited By ~ 3

Author(s):

O. L. V. Costa ◽

F. Dufour

Keyword(s):

Markov Process ◽

Probability Measure ◽

Continuous Time ◽

Invariant Probability Measure ◽

Probability Measures ◽

Transition Kernel ◽

Occupation Measure ◽

Ergodic Properties ◽

Necessary And Sufficient ◽

Separable Metric Space

In this paper we obtain some ergodic properties and ergodic decompositions of a continuous-time, Borel right Markov process taking values in a locally compact and separable metric space. Initially, we assume that an invariant probability measure (IPM) μ exists for the process and, without making any further assumptions on the transition kernel, obtain some characterization results for the convergence of the expected occupation measure to a limit kernel. Under the same assumption, we present the so-called Yosida decomposition. Next, instead of assuming the existence of an IPM, we assume that the Markov process satisfies a certain condition, named the T'-condition. Under this condition it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an IPM and that the process admits a Doeblin decomposition. Furthermore, it is shown that in this case the set of ergodic probability measures is countable and that every probability measure for the Markov process is nonsingular with respect to the transition kernel.

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Ergodic properties and ergodic decompositions of continuous-time Markov processes

Journal of Applied Probability ◽

10.1017/s0021900200002096 ◽

2006 ◽

Vol 43 (03) ◽

pp. 767-781

Author(s):

O. L. V. Costa ◽

F. Dufour

Keyword(s):

Markov Process ◽

Probability Measure ◽

Continuous Time ◽

Invariant Probability Measure ◽

Probability Measures ◽

Transition Kernel ◽

Occupation Measure ◽

Ergodic Properties ◽

Necessary And Sufficient ◽

Separable Metric Space

In this paper we obtain some ergodic properties and ergodic decompositions of a continuous-time, Borel right Markov process taking values in a locally compact and separable metric space. Initially, we assume that an invariant probability measure (IPM) μ exists for the process and, without making any further assumptions on the transition kernel, obtain some characterization results for the convergence of the expected occupation measure to a limit kernel. Under the same assumption, we present the so-called Yosida decomposition. Next, instead of assuming the existence of an IPM, we assume that the Markov process satisfies a certain condition, named theT'-condition. Under this condition it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an IPM and that the process admits a Doeblin decomposition. Furthermore, it is shown that in this case the set of ergodic probability measures is countable and that every probability measure for the Markov process is nonsingular with respect to the transition kernel.

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On uniformly distributed orbits of certain horocycle flows

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700001474 ◽

1982 ◽

Vol 2 (2) ◽

pp. 139-158 ◽

Cited By ~ 18

Author(s):

S. G. Dani

Keyword(s):

Probability Measure ◽

Periodic Point ◽

Open Subset ◽

Invariant Probability Measure ◽

Zero Measure ◽

Lebesque Measure ◽

Image Position

AbstractLet(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

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Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

ESAIM Probability and Statistics ◽

10.1051/ps/2019007 ◽

2019 ◽

Vol 23 ◽

pp. 797-802

Author(s):

Raphaël Cerf ◽

Joseba Dalmau

Keyword(s):

Markov Chain ◽

Probability Measure ◽

Branching Process ◽

Invariant Probability Measure ◽

Classical Formula ◽

Multitype Branching Process ◽

Primitive Matrix ◽

Watson Process

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

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Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

Canadian Mathematical Bulletin ◽

10.4153/cmb-2015-025-7 ◽

2015 ◽

Vol 58 (3) ◽

pp. 471-485 ◽

Cited By ~ 1

Author(s):

Seckin Demirbas

Keyword(s):

Probability Measure ◽

Initial Data ◽

Schrödinger Equation ◽

Schrodinger Equation ◽

Invariant Probability Measure ◽

Well Posedness ◽

Cubic Nonlinearity ◽

Cubic Schrödinger Equation ◽

Well Posed ◽

Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

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