ergodic properties of color records abstract in a series of papers culminating in the dissertation of the second author under the direction of p w kasteleyn several rigorous results have been obtained concerning the color sequence seen by a random walker on a stochastically black white colored lattice and this under the sole assumption that the coloring is stationary ergodic and independent of the walking in this article we investigate the ergodic properties of the color record process and their implications for the behavior of the sequence nk k 0 of successive times needed for the walker to go from the kth to the k 1 st black point as an example of the type of results obtained we show that if a certain induced color record process is strongly mixing in the ergodic theoretic sense then the average nk tends to the inverse of the density of black points as k tends to infinity and similarly that the distribution of nk tends to a limit that can be explicitly calculated in terms of the distribution of n0 for a certain class of stochastic colorings and random walks we are able to verify that the induced color record process is indeed strongly mixing our main goal however is to clarify the nature of the ergodic properties and thereby provide a tool for future use in the study of color records

locally linear reconstruction for instance based learning instance based learning ibl so called memory based reasoning mbr is a commonly used non parametric learning algorithm k nearest neighbor k nn learning is the most popular realization of ibl due to its usability and adaptability k nn has been successfully applied to a wide range of applications however in practice one has to set important model parameters only empirically the number of neighbors k and weights to those neighbors in this paper we propose structured ways to set these parameters based on locally linear reconstruction llr we then employed sequential minimal optimization smo for solving quadratic programming step involved in llr for classification to reduce the computational complexity experimental results from 11 classification and eight regression tasks were promising enough to merit further investigation not only did llr outperform the conventional weight allocation methods without much additional computational cost but also llr was found to be robust to the change of k

active learning based on minimization of the expected path length of random walks on the learned manifold structure abstract active learning algorithms aim at selecting important samples to label for subsequent machine learning tasks many active learning algorithms make use of the reproducing kernel hilbert space rkhs induced by a gaussian radial basis function rbf kernel and leverage the geometrical structure of the data for query sample selection parameters for the kernel function and the k nearest neighborhood graph must be properly set beforehand as a tool exploring the structure of data active learning algorithms with automatic tuning of those parameters are desirable in this paper local linear embedding lle with convex constraints on neighbor weights is used to learn the geometrical structure of the data in the rkhs induced by a gaussian rbf kernel automatic tuning of the kernel parameter is based on the assumption that the geometrical structure of the data in the rkhs is sparse and local with the markov matrix established based on the learned lle weight matrix the total expected path length of the random walks from all samples to selected samples is proposed to be a criterion for query sample selection a greedy algorithm having a guaranteed solution bound is developed to select query samples and a two phase scheme is also proposed for scaling the proposed active learning algorithm experimental results on data sets including hundreds to tens of thousands of samples have shown the feasibility of the proposed approach

dynamical spin system exact solution and mean recurrence time abstract a model involving a chain of n 2 spins si 1 i 1 n evolvi ng syncronously in discrete time t via a nonlinear autonomous transformation si t 1 si t si 1 t i 1 n 1 sn t 1 sn t is presented the transformation equations are solved explicitly and the detailed decomposition of state space into ergodic sets is found on the assumption of equally likely initial states the mean recurrence time is calculated and its variance is discussed the model displays a strikingly sensitive dependence on the number of spins and this is reflected in the staircase behavior of the mean recurrence time remarks are made regarding the connection between the behavior of the model and the ground states of a related two dimensional ising model

piecewise automatic groups we introduce a notion of a piecewise automatic group among these groups we describe a new class of groups of intermediate growth we show that for any function f n n there exists a finitely generated torsion group of intermediate growth g for which the folner function satisfies folg s n f n for some generating set s and all sufficiently large n as a corollary we see that the asymptotic entropy of simple random walks on these groups could be arbitrarily close to being linear while the poisson boundary is trivial

variations on a theme by mark kac mark kac s theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations most of which are formulated in terms of 0 1 processes whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0 this condition is dropped in part of the variations two others refer to the variance of the recurrence time and two to the poincare cycle of a dynamical system most variations consist in inequalities and formal identities for the mean first arrival time and subsequent recurrence times for the given state

heat kernel upper bounds for jump processes and the first exit time for all t 0 x m and f l2 m the function pt x y can be considered as the transition density of the associated markov processx xt t 0 and the question of estimating of pt x y which is the main subject of this paper is closely related to the properties of x the function pt x y is also referred to as a heat kernel and this terminology goes back to the classical gauss weierstrass heat kernel associated with the heat semigroup e t 0 in r n whose markov process is brownian motion a somewhat more general but still well treated case is when m d is a riemannian metric measure space that is m is a riemannian manifold

poisson furstenberg boundary and growth of groups we study the poisson furstenberg boundary of random walks on permutational wreath products we give a sufficient condition for a group to admit a symmetric measure of finite first moment with non trivial boundary and show that this criterion is useful to establish exponential word growth of groups we construct groups of exponential growth such that all finitely supported not necessarily symmetric possibly degenerate random walks on these groups have trivial boundary this gives a negative answer to a question of kaimanovich and vershik

active learning based on locally linear reconstruction we consider the active learning problem which aims to select the most representative points out of many existing active learning techniques optimum experimental design oed has received considerable attention recently the typical oed criteria minimize the variance of the parameter estimates or predicted value however these methods see only global euclidean structure while the local manifold structure is ignored for example i optimal design selects those data points such that other data points can be best approximated by linear combinations of all the selected points in this paper we propose a novel active learning algorithm which takes into account the local structure of the data space that is each data point should be approximated by the linear combination of only its neighbors given the local reconstruction coefficients for every data point and the coordinates of the selected points a transductive learning algorithm called locally linear reconstruction llr is proposed to reconstruct every other point the most representative points are thus defined as those whose coordinates can be used to best reconstruct the whole data set the sequential and convex optimization schemes are also introduced to solve the optimization problem the experimental results have demonstrated the effectiveness of our proposed method

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

mixing properties for random walk in random scenery consider the lattice z d d geq 1 together with a stochastic black white coloring of its points and on it a random walk that is independent of the coloring a local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position under weak assumptions on the probability distributions governing walk and coloring we prove asymptotic independence of local sceneries perceived at times 0 and n in the limit as n rightarrow infty and at times 0 and t k in the limit as k rightarrow infty where t k is the random k th hitting time of a black point an immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits i e of t k 1 t k as k rightarrow infty the limit distribution is expressed in terms of the distribution of the first hitting time t 1 the proof uses coupling arguments and ergodic theory

heat kernel estimates for jump processes of mixed types on metric measure spaces in this paper we investigate symmetric jump type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces the class of metric measure spaces includes the alfors d regular sets which is a class of fractal sets that contains geometrically self similar sets a typical example of our jump type processes is the symmetric jump process with jumping intensity e c 0 x y x y int alpha 1 alpha 2 frac c alpha x y x y d alpha nu d alpha where is a probability measure on alpha 1 alpha 2 subset 0 2 c x y is a jointly measurable function that is symmetric in x y and is bounded between two positive constants and c 0 x y is a jointly measurable function that is symmetric in x y and is bounded between 1 and 2 where either 2 1 0 or 1 2 0 this example contains mixed symmetric stable processes on mathbb r n as well as mixed relativistic symmetric stable processes on mathbb r n we establish parabolic harnack principle and derive sharp two sided heat kernel estimate for such jump type processes

growth of periodic grigorchuk groups on torsion grigorchuk groups we construct random walks of finite entropy and power law tail decay with non trivial poisson boundary such random walks provide near optimal volume lower estimates for these groups in particular for the first grigorchuk group g we show that its growth v g s n satisfies lim n rightarrow infty log log v g s n log n alpha 0 where alpha 0 frac log 2 log lambda 0 approx 0 7674 lambda 0 is the positive root of the polynomial x 3 x 2 2x 4

growth of permutational extensions we study the geometry of a class of group extensions containing permutational wreath products which we call permutational extensions we construct for all k a torsion group kk with growth function v k k n sim exp n 1 1 alpha k qquad 2 3 3 alpha 2 2 2 alpha 2 1 1 alpha 2 and a torsion free group hk with growth function v h k n sim exp log n n 1 1 alpha k these are the first examples of groups of intermediate growth for which the asymptotics of their growth function is known

explicit constants for gaussian upper bounds on heat kernels on utilise des inegalites logarithmiques de sobolev pour obtenir des bornes superieures gaussiennes sur les noyaux de la chaleur associes a des operateurs d ordre 2 elliptiques et hypoelliptiques

random walks on lattices with points of two colours i abstract this paper is concerned with random walks on lattices with two kinds of points black and white the colours of the points are random variables with a translation invariant but otherwise arbitrary joint probability distribution the steps of the walk are independent of the colours we study the stochastic properties of the length of the subwalk from the starting point to a first black point and of subwalks between points visited in succession and establish a number of exact relations these relations can be applied to a trapping problem by identifying the black points with imperfect traps an example is discussed

boundary behavior for groups of subexponential growth in this paper we introduce a method for partial description of the poisson boundary for a certain class of groups acting on a segment as an application we find among the groups of subexponential growth those that admit nonconstant bounded harmonic functions with respect to some symmetric infinitely supported measure of finite entropy h this implies that the entropy h of the corresponding random walk is finite and positive as another application we exhibit certain discontinuity for the recurrence property of random walks finally as a corollary of our results we get new estimates from below for the growth function of a certain class of grigorchuk groups in particular we exhibit the first example of a group generated by a finite state automaton such that the growth function is subexponential but grows faster than exp n for any 1 we show that in some of our examples the growth function satisfies exp n ln

optimising kernel parameters and regularisation coefficients for non linear discriminant analysis in this paper we consider a novel bayesian interpretation of fisher s discriminant analysis we relate rayleigh s coefficient to a noise model that minimises a cost based on the most probable class centres and that abandons the regression to the labels assumption used by other algorithms optimisation of the noise model yields a direction of discrimination equivalent to fisher s discriminant and with the incorporation of a prior we can apply bayes rule to infer the posterior distribution of the direction of discrimination nonetheless we argue that an additional constraining distribution has to be included if sensible results are to be obtained going further with the use of a gaussian process prior we show the equivalence of our model to a regularised kernel fisher s discriminant a key advantage of our approach is the facility to determine kernel parameters and the regularisation coefficient through the optimisation of the marginal log likelihood of the data an added bonus of the new formulation is that it enables us to link the regularisation coefficient with the generalisation error

bounds on the spectrum for markov chains and markov processes a generalization of cheeger s inequality we prove a general version of cheeger s inequality for discrete time markov chains and continuous time markovian jump processes both reversible and nonreversible with general state space we also prove a version of cheeger s inequality for markov chains and processes with killing as an application we prove l 2 exponential convergence to equilibrium for random walk with inward drift on a class of countable rooted graphs