kernel methods in machine learning we review machine learning methods employing positive definite kernels these methods formulate learning and estimation problems in a reproducing kernel hilbert space rkhs of functions defined on the data domain expanded in terms of a kernel working in linear spaces of function has the benefit of facilitating the construction and analysis of learning algorithms while at the same time allowing large classes of functions the latter include nonlinear functions as well as functions defined on nonvectorial data we cover a wide range of methods ranging from binary classifiers to sophisticated methods for estimation with structured data

a law of large numbers for identically distributed martingale differences the averages of an identically distributed martingale difference sequence converge in mean to zero but the almost sure convergence of the averages characterizes l log l in the following sense if the terms of an identically distributed martingale difference sequence are in l log l the averages converge to zero almost surely but if f is any integrable random variable with zero expectation which is not in l log l there is a martingale difference sequence whose terms have the same distributions as f and whose averages diverge almost surely the maximal function of the averages of an identically distributed martingale difference sequence is integrable if its terms are in l log l the converse is false

characteristic kernels and infinitely divisible distributions we connect shift invariant characteristic kernels to infinitely divisible distributions on rd characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures the contribution of this paper is twofold first we show using the levy khintchine formula that any shift invariant kernel given by a bounded continuous and symmetric probability density function pdf of an infinitely divisible distribution on rd is characteristic we mention some closure properties of such characteristic kernels under addition pointwise product and convolution second in developing various kernel mean algorithms it is fundamental to compute the following values i kernel mean values mp x x and ii kernel mean rkhs inner products mp mq h for probability measures p q if p q and kernel k are gaussians then the computation of i and ii results in gaussian pdfs that are tractable we generalize this gaussian combination to more general cases in the class of infinitely divisible distributions we then introduce a conjugate kernel and a convolution trick so that the above i and ii have the same pdf form expecting tractable computation at least in some cases as specific instances we explore stable distributions and a rich class of generalized hyperbolic distributions where the laplace cauchy and student s t distributions are included

trajectory stabilization of a model car via fuzzy control abstract this paper deals with trajectory stabilization of a computer simulated model car via fuzzy control stability conditions of fuzzy systems are given in accordance with the definition of stability in the sense of lyapunov first we approximate a computer simulated model car whose dynamics is nonlinear by t s takagi and sugeno fuzzy model fuzzy control rules which guarantee stability of the control system under a condition are derived from the approximated fuzzy model the simulation results show that the fuzzy control rules effectively realize trajectory stabilization of the model car along a given reference trajectory from all initial positions under a condition and the dynamics of the approximated fuzzy model agrees well with that of the model car

discrete time controlled markov processes with average cost criterion a survey this work is a survey of the average cost control problem for discrete time markov processes the authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades the exposition ranges from finite to borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies the authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography the authors have also identified several important questions that are still open to investigation

a kernel two sample test we propose a framework for analyzing and comparing distributions which we use to construct statistical tests to determine if two samples are drawn from different distributions our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel hilbert space rkhs and is called the maximum mean discrepancy mmd we present two distribution free tests based on large deviation bounds for the mmd and a third test based on the asymptotic distribution of this statistic the mmd can be computed in quadratic time although efficient linear time approximations are available our statistic is an instance of an integral probability metric and various classical metrics on distributions are obtained when alternative function classes are used in place of an rkhs we apply our two sample tests to a variety of problems including attribute matching for databases using the hungarian marriage method where they perform strongly excellent performance is also obtained when comparing distributions over graphs for which these are the first such tests

interval type 2 fuzzy model predictive control of nonlinear networked control systems in this paper the problem of fuzzy predictive control of nonlinear networked control systems subject to parameter uncertainties and defective communication links is studied stochastic variables with bernoulli random binary distribution are used to represent the defective communication links with packets loss occurring intermittently between the controller and the physical plant an interval type 2 it2 takagi sugeno t s fuzzy model is employed to describe the nonlinear plant subject to parameter uncertainties which can be captured with the lower and upper membership functions the it2 fuzzy model and it2 fuzzy controller are not required to share the same lower and upper membership functions in order to design the state feedback fuzzy model predictive controller an optimization problem which minimizes the upper bound of a quadratic objective function subject to input constraints and packets dropout is formulated and solved at every sampling instant in the finite time horizon by introducing some slack matrices less conservative conditions are developed for system stability analysis two examples are given to demonstrate the effectiveness and merits of the proposed new design techniques

simplified interval type 2 fuzzy logic systems type reduction tr followed by defuzzification is commonly used in interval type 2 fuzzy logic systems it2 flss because of the iterative nature of tr it may be a computational bottleneck for the real time applications of an it2 fls this has led to many direct approaches to defuzzification that bypass tr the simplest of which is the nie tan direct defuzzification method nt method this paper provides some theoretical analyses of the nt method that answer the question why is the nt method good to use this paper also provides a direct relationship between tr followed by defuzzification using km algorithms and the nt method it also provides an improved nt method numerical examples illustrate our theoretical results and suggest that the nt method is a very good way to simplify an interval type 2 fuzzy set

an alternative method of cross validation for the smoothing of density estimates cross validation with kullback leibler loss function has been applied to the choice of a smoothing parameter in the kernel method of density estimation a framework for this problem is constructed and used to derive an alternative method of cross validation based on integrated squared error recently also proposed by rudemo 1982 hall 1983 has established the consistency and asymptotic optimality of the new method for small and moderate sized samples the performances of the two methods of cross validation are compared on simulated data and specific examples

kernel bayes rule bayesian inference with positive definite kernels a kernel method for realizing bayes rule is proposed based on representations of probabilities in reproducing kernel hilbert spaces probabilities are uniquely characterized by the mean of the canonical map to the rkhs the prior and conditional probabilities are expressed in terms of rkhs functions of an empirical sample no explicit parametric model is needed for these quantities the posterior is likewise an rkhs mean of a weighted sample the estimator for the expectation of a function of the posterior is derived and rates of consistency are shown some representative applications of the kernel bayes rule are presented including bayesian computation without likelihood and filtering with a nonparametric state space model

universality characteristic kernels and rkhs embedding of measures over the last few years two different notions of positive definite pd kernels universal and characteristic have been developing in parallel in machine learning universal kernels are proposed in the context of achieving the bayes risk by kernel based classification regression algorithms while characteristic kernels are introduced in the context of distinguishing probability measures by embedding them into a reproducing kernel hilbert space rkhs however the relation between these two notions is not well understood the main contribution of this paper is to clarify the relation between universal and characteristic kernels by presenting a unifying study relating them to rkhs embedding of measures in addition to clarifying their relation to other common notions of strictly pd conditionally strictly pd and integrally strictly pd kernels for radial kernels on d all these notions are shown to be equivalent

unsupervised state space modeling using reproducing kernels a novel framework for the design of state space models ssms is proposed whereby the state transition function of the model is parametrized using reproducing kernels the nature of ssms requires learning a latent function that resides in the state space and for which input output sample pairs are not available thus prohibiting the use of gradient based supervised kernel learning to this end we then propose to learn the mixing weights of the kernel estimate by sampling from their posterior density using monte carlo methods we first introduce an offline version of the proposed algorithm followed by an online version which performs inference on both the parameters and the hidden state through particle filtering the accuracy of the estimation of the state transition function is first validated on synthetic data next we show that the proposed algorithm outperforms kernel adaptive filters in the prediction of real world time series while also providing probabilistic estimates a key advantage over standard methods

filtering with state observation examples via kernel monte carlo filter this letter addresses the problem of filtering with a state space model standard approaches for filtering assume that a probabilistic model for observations i e the observation model is given explicitly or at least parametrically we consider a setting where this assumption is not satisfied we assume that the knowledge of the observation model is provided only by examples of state observation pairs this setting is important and appears when state variables are defined as quantities that are very different from the observations we propose kernel monte carlo filter a novel filtering method that is focused on this setting our approach is based on the framework of kernel mean embeddings which enables nonparametric posterior inference using the state observation examples the proposed method represents state distributions as weighted samples propagates these samples by sampling estimates the state posteriors by kernel bayes rule and resamples by kernel herding in particular the sampling and resampling procedures are novel in being expressed using kernel mean embeddings so we theoretically analyze their behaviors we reveal the following properties which are similar to those of corresponding procedures in particle methods the performance of sampling can degrade if the effective sample size of a weighted sample is small and resampling improves the sampling performance by increasing the effective sample size we first demonstrate these theoretical findings by synthetic experiments then we show the effectiveness of the proposed filter by artificial and real data experiments which include vision based mobile robot localization

some new results on sample path optimality in ergodic control of diffusions we present some new results on sample path optimality for the ergodic control problem of a class of nondegenerate diffusions controlled through the drift the hypothesis most often used in the literature to ensure the existence of an almost sure sample path optimal stationary markov control requires finite second moments of the first hitting times tau of bounded domains over all admissible controls we show that this can be considerably weakened mathbb e tau 2 may be replaced with mathbb e tau ln tau thus reducing the required rate of convergence of averages from polynomial to logarithmic a foster lyapunov condition that guarantees this is also exhibited moreover we study a large class of models that are neither uniformly stable nor have a near monotone running cost and we exhibit sufficient conditions for the existence of a sample path optimal stationary markov control

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

ergodic control of multidimensional diffusions 1 the existence results the existence of optimal stable markov relaxed controls for the ergodic control of multidimensional diffusions is established by direct probabilistic methods based on a characterization of a s limit sets of empirical measures the optimality of the above is established in the strong i e almost sure sense among all admissible controls under very general conditions

sample path optimality and variance minimization of average cost markov control processes this paper studies several average cost criteria for markov control processes on borel spaces with possibly unbounded costs under suitable hypotheses we show i the existence of a sample path average cost spac optimal stationary policy ii a stationary policy is spac optimal if and only if it is expected average cost eac optimal and iii within the class of stationary spac optimal equivalently eac optimal policies there exists one with a minimal limiting average variance

control of markov chains with long run average cost criterion the dynamic programming equations the long run average cost control problem for discrete time markov chains on a countable state space is studied in a very general framework necessary and sufficient conditions for optimality in terms of the dynamic programming equations are given when an optimal stable stationary strategy is known to exist e g for the situations studied in stochastic differential systems stochastic control theory and applications ima vol math app 10 springer verlag new york berlin 1988 pp 57 77 a characterization of the desired solution of the dynamic programming equations is given in a special case also included is a novel convex analytic argument for deducing the existence of an optimal stable stationary strategy when that of a randomized one is known

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

filter design for interval type 2 fuzzy systems with d stability constraints under a unified frame this paper investigates the problem of filter design for interval type 2 it2 fuzzy systems with d stability constraints based on a new performance index attention is focused on solving the h infty l 2 l infty passive and dissipativity fuzzy filter design problems for it2 fuzzy systems with d stability constraints in a unified frame under the new performance index frame using lyapunov stability theory a novel type of it2 filter is designed such that the filtering error system guarantees the prescribed h infty l 2 l infty passive and dissipativity performance levels with d stability constraints the existence condition of the it2 filter is expressed as the convex optimization problem and the filter parameters in the condition can be solved by the standard software the it2 fuzzy model and it2 fuzzy filter do not need to share the same lower and upper membership functions finally a numerical example is provided to show the effectiveness of the proposed results

kernel dimension reduction in regression we present a new methodology for sufficient dimension reduction sdr our methodology derives directly from the formulation of sdr in terms of the conditional independence of the covariate x from the response y given the projection of x on the central subspace cf j amer statist assoc 86 1991 316 342 and regression graphics 1998 wiley we show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel hilbert spaces and we show how this characterization leads to an m estimator for the central subspace the resulting estimator is shown to be consistent under weak conditions in particular we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for sdr methods we also present empirical results showing that the new methodology is competitive in practice

efficient svm training using low rank kernel representations svm training is a convex optimization problem which scales with the training set size rather than the feature space dimension while this is usually considered to be a desired quality in large scale problems it may cause training to be impractical the common techniques to handle this difficulty basically build a solution by solving a sequence of small scale subproblems our current effort is concentrated on the rank of the kernel matrix as a source for further enhancement of the training procedure we first show that for a low rank kernel matrix it is possible to design a better interior point method ipm in terms of storage requirements as well as computational complexity we then suggest an efficient use of a known factorization technique to approximate a given kernel matrix by a low rank matrix which in turn will be used to feed the optimizer finally we derive an upper bound on the change in the objective function value based on the approximation error and the number of active constraints support vectors this bound is general in the sense that it holds regardless of the approximation method

general type 2 fuzzy logic systems made simple a tutorial the purpose of this tutorial paper is to make general type 2 fuzzy logic systems gt2 flss more accessible to fuzzy logic researchers and practitioners and to expedite their research designs and use to accomplish this the paper 1 explains four different mathematical representations for general type 2 fuzzy sets gt2 fss 2 demonstrates that for the optimal design of a gt2 fls one should use the vertical slice representation of its gt2 fss because it is the only one of the four mathematical representations that is parsimonious 3 shows how to obtain set theoretic and other operations for gt2 fss using type 1 t1 fs mathematics cuts play a central role 4 reviews mamdani and tsk interval type 2 it2 flss so that their mathematical operations can be easily used in a gt2 fls 5 provides all of the formulas that describe both mamdani and tsk gt2 flss 6 explains why center of sets type reduction should be favored for a gt2 fls over centroid type reduction 7 provides three simplified gt2 flss two are for mamdani gt2 flss and one is for a tsk gt2 fls all of which bypass type reduction and are generalizations from their it2 fls counterparts to gt2 flss 8 explains why gradient based optimization should not be used to optimally design a gt2 fls 9 explains how derivative free optimization algorithms can be used to optimally design a gt2 fls and 10 provides a three step approach for optimally designing flss in a progressive manner from t1 to it2 to gt2 each of which uses a quantum particle swarm optimization algorithm by virtue of which the performance for the it2 fls cannot be worse than that of the t1 fls and the performance for the gt2 fls cannot be worse than that of the it2 fls

a note on the rate of convergence in the strong law of large numbers for martingales abstract we prove a baum katz nagaev type rate of convergence in the marcinkiewicz zygmund and kolmogorov strong laws of large numbers for norm bounded martingale difference sequences

on learning vector valued functions in this letter we provide a study of learning in a hilbert space of vectorvalued functions we motivate the need for extending learning theory of scalar valued functions by practical considerations and establish some basic results for learning vector valued functions that should prove useful in applications specifically we allow an output space y to be a hilbert space and we consider a reproducing kernel hilbert space of functions whose values lie in y in this setting we derive the form of the minimal norm interpolant to a finite set of data and apply it to study some regularization functionals that are important in learning theory we consider specific examples of such functionals corresponding to multiple output regularization networks and support vector machines for both regression and classification finally we provide classes of operator valued kernels of the dot product and translation invariant type

sample path average optimality for markov control processes the authors consider a markov control process with borel state and actions spaces unbounded costs and under the long run sample path average cost criterion they prove that under very weak assumptions on the transition law and a moment assumption for the one step cost there exists a stationary policy with invariant probability distribution v that is sample path average cost optimal for v almost all initial states in addition every expected average cost optimal stationary policy is in fact liminf sample path average cost optimal and strongly expected average cost optimal

query efficient posterior estimation in scientific experiments via bayesian active learning a common problem in disciplines of applied statistics research such as astrostatistics is of estimating the posterior distribution of relevant parameters typically the likelihoods for such models are computed via expensive experiments such as cosmological simulations of the universe an urgent challenge in these research domains is to develop methods that can estimate the posterior with few likelihood evaluations in this paper we study active posterior estimation in a bayesian setting when the likelihood is expensive to evaluate existing techniques for posterior estimation are based on generating samples representative of the posterior such methods do not consider efficiency in terms of likelihood evaluations in order to be query efficient we treat posterior estimation in an active regression framework we propose two myopic query strategies to choose where to evaluate the likelihood and implement them using gaussian processes via experiments on a series of synthetic and real examples we demonstrate that our approach is significantly more query efficient than existing techniques and other heuristics for posterior estimation

hilbert space embeddings and metrics on probability measures a hilbert space embedding for probability measures has recently been proposed with applications including dimensionality reduction homogeneity testing and independence testing this embedding represents any probability measure as a mean element in a reproducing kernel hilbert space rkhs a pseudometric on the space of probability measures can be defined as the distance between distribution embeddings we denote this as k indexed by the kernel function k that defines the inner product in the rkhs we present three theoretical properties of k first we consider the question of determining the conditions on the kernel k for which k is a metric such k are denoted characteristic kernels unlike pseudometrics a metric is zero only when two distributions coincide thus ensuring the rkhs embedding maps all distributions uniquely i e the embedding is injective while previously published conditions may apply only in restricted circumstances e g on compact domains and are difficult to check our conditions are straightforward and intuitive integrally strictly positive definite kernels are characteristic alternatively if a bounded continuous kernel is translation invariant on d then it is characteristic if and only if the support of its fourier transform is the entire d second we show that the distance between distributions under k results from an interplay between the properties of the kernel and the distributions by demonstrating that distributions are close in the embedding space when their differences occur at higher frequencies third to understand the nature of the topology induced by k we relate k to other popular metrics on probability measures and present conditions on the kernel k under which k metrizes the weak topology

on the influence of the kernel on the consistency of support vector machines in this article we study the generalization abilities of several classifiers of support vector machine svm type using a certain class of kernels that we call universal it is shown that the soft margin algorithms with universal kernels are consistent for a large class of classification problems including some kind of noisy tasks provided that the regularization parameter is chosen well in particular we derive a simple sufficient condition for this parameter in the case of gaussian rbf kernels on the one hand our considerations are based on an investigation of an approximation property the so called universality of the used kernels that ensures that all continuous functions can be approximated by certain kernel expressions this approximation property also gives a new insight into the role of kernels in these and other algorithms on the other hand the results are achieved by a precise study of the underlying optimization problems of the classifiers furthermore we show consistency for the maximal margin classifier as well as for the soft margin svm s in the presence of large margins in this case it turns out that also constant regularization parameters ensure consistency for the soft margin svm s finally we prove that even for simple noise free classification problems svm s with polynomial kernels can behave arbitrarily badly