support vector machine soft margin classifiers error analysis the purpose of this paper is to provide a pac error analysis for the q norm soft margin classifier a support vector machine classification algorithm it consists of two parts regularization error and sample error while many techniques are available for treating the sample error much less is known for the regularization error and the corresponding approximation error for reproducing kernel hilbert spaces we are mainly concerned about the regularization error it is estimated for general distributions by a k functional in weighted lq spaces for weakly separable distributions i e the margin may be zero satisfactory convergence rates are provided by means of separating functions a projection operator is introduced which leads to better sample error estimates especially for small complexity kernels the misclassification error is bounded by the v risk associated with a general class of loss functions v the difficulty of bounding the offset is overcome polynomial kernels and gaussian kernels are used to demonstrate the main results the choice of the regularization parameter plays an important role in our analysis

time regularity for random walks on locally compact groups let g be a compactly generated locally compact group and let t be the operator of convolution with a probability measure on g our main results give sufficient conditions on for the operator t to be analytic in l p g 1 p where analyticity means that one has an estimate of form vert i t t n vert leq cn 1 for all n 1 2 in l p operator norm counterexamples show that analyticity may not hold if some of the conditions are not satisfied

heat kernel estimates and riesz transforms on some riemannian covering manifolds consider a riemannian manifold m which is a galois covering of a compact manifold with nilpotent deck transformation group g for the laplace operator on m we prove a precise estimate for the gradient of the heat kernel and show that the riesz transforms are bounded in l p m 1 p we also obtain estimates for discrete oscillations of the heat kernel and boundedness of discrete riesz transform operators which are defined using the action of g on m

fast rates for support vector machines using gaussian kernels for binary classification we establish learning rates up to the order of n 1 for support vector machines svms with hinge loss and gaussian rbf kernels these rates are in terms of two assumptions on the considered distributions tsybakov s noise assumption to establish a small estimation error and a new geometric noise condition which is used to bound the approximation error unlike previously proposed concepts for bounding the approximation error the geometric noise assumption does not employ any smoothness assumption 1 introduction in recent years support vector machines svms have been the subject of many theoretical considerations despite this effort their learning performance on restricted classes of distributions is still widely unknown in particular it is unknown under which nontrivial circumstances svms can guarantee fast learning rates the aim of this work is to use concepts like tsybakov s noise assumption and local rademacher averages to establish learning rates up to the order of n 1 for nontrivial distributions in addition to these concepts that are used to deal with the stochastic part of the analysis we also introduce a geometric assumption for distributions that allows us to estimate the approximation properties of gaussian rbf kernels unlike many other concepts introduced for bounding the approximation error our geometric assumption is not in terms of smoothness but describes the concentration and the noisiness of the data generating distribution near the decision boundary let us formally introduce the statistical classification problem to this end let us fix a subset x r d we write y 1 1 given a finite training set

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

polynomial growth harmonic functions on groups of polynomial volume growth we consider harmonic functions of polynomial growth of some order d on cayley graphs of discrete groups of polynomial volume growth of order d w r t the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions more precisely the dimension of this space of harmonic functions is at most of order d d 1 as in the already known riemannian case this estimate is polynomial in the growth degree more generally our techniques also apply to graphs roughly isometric to cayley graphs of groups of polynomial volume growth

random walks on discrete groups of polynomial volume growth let mu be a probability measure with finite support on a discrete group gamma of polynomial volume growth the main purpose of this paper is to study the asymptotic behavior of the convolution powers mu n of mu if mu is centered then we prove upper and lower gaussian estimates we prove a central limit theorem and we give a generalization of the berry esseen theorem these results also extend to noncentered probability measures we study the associated riesz transform operators the main tool is a parabolic harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of krylov and safonov this inequality implies that the positive mu harmonic functions are constant finally we give a characterization of the mu harmonic functions which grow polynomially

bi level path following for cross validated solution of kernel quantile regression we show how to follow the path of cross validated solutions to families of regularized optimization problems defined by a combination of a parameterized loss function and a regularization term a primary example is kernel quantile regression where the parameter of the loss function is the quantile being estimated even though the bi level optimization problem we encounter for every quantile is non convex the manner in which the optimal cross validated solution evolves with the parameter of the loss function allows tracking of this solution we prove this property construct the resulting algorithm and demonstrate it on real and artificial data this algorithm allows us to efficiently solve the whole family of bi level problems we show how it can be extended to cover other modeling problems like support vector regression and alternative in sample model selection approaches

dimension free estimates for discrete riesz transforms on products of abelian groups let g be a lca group with a fixed g0 g spanning an infinite subgroup let j acting on l2 gn be translation by go in the jth coordinate the discrete derivatives j i j define a discrete laplacian l j 1n j j and discrete riesz transforms rj jl 12 we get dimension free estimates cp f lp gn j 1n rj f 2 rj f 212lp gn cp f lp gn for 2 p the right side inequality is false for 1 p 2 but we give the substitute we also consider the case where gpg0 is a cyclic group

a finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment we consider a nearest neighbor symmetric random walk on a homogeneous ergodic random lattice z d the jump rates of the walk are independent identically bernoulli distributed random variables indexed by the bonds of the lattice a standard result from the homogenization theory see a de masi p a ferrari s goldstein w d wick an invariance principle for reversible markov processes applications to random walks in random environments j statist phys 55 3 4 1989 787 855 asserts that the scaled trajectory of the particle satisfies the functional central limit theorem the covariance matrix of the limiting normal distribution is called the effective diffusivity of the walk we use the duality structure corresponding to the product bernoulli measure to construct a numerical scheme that approximates this parameter when d 3 the estimates of the convergence rates are also provided

free knot polynomial splines with confidence intervals the estimates of knot locations and coefficients are obtained through a non linear least squares solution that corresponds to the maximum likelihood estimate confidence intervals are then constructed based on the asymptotic distribution of the maximum likelihood estimator average coverage probabilities and the accuracy of the estimate are examined via simulation this includes comparisons between our method and some existing methods such as smoothing spline and variable knots selection as well as a bayesian version of the variable knots method simulation results indicate that our method works well for smooth underlying functions and also reasonably well for discontinuous functions it also performs well for fairly small sample sizes

transition probabilities for symmetric jump processes we consider symmetric markov chains on the integer lattice in d dimensions where a 0 2 and the conductance between x and y is comparable to x y d we establish upper and lower bounds for the transition probabilities that are sharp up to constants

local limit theorem for symmetric random walks in gromov hyperbolic groups completing a strategy of gouezel and lalley we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non elementary gromov hyperbolic group denoting by r the inverse of the spectral radius of the random walk the probability to return to the identity at time n behaves like c r n n 3 2 an important step in the proof is to extend ancona s results on the martin boundary up to the spectral radius we show that the martin boundary for r harmonic functions coincides with the geometric boundary of the group in an appendix we explain how the symmetry assumption of the measure can be dispensed with for surface groups

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

on a neumann problem with critical exponents and hardy potentials abstract in this paper we consider the neumann problem of the potential elliptic system with hardy terms and critical exponents on a bounded domain using the critical point theory we first find infinitely many solutions for the corresponding perturbed subcritical problem analyzing concentration points in the bubbles carefully which appear in the approximate solutions we establish the concentration estimates and then obtain infinitely many solutions with positive energy by passing to a limit

statistical performance of support vector machines the support vector machine svm algorithm is well known to the computer learning community for its very good practical results the goal of the present paper is to study this algorithm from a statistical perspective using tools of concentration theory and empirical processes our main result builds on the observation made by other authors that the svm can be viewed as a statistical regularization procedure from this point of view it can also be interpreted as a model selection principle using a penalized criterion it is then possible to adapt general methods related to model selection in this framework to study two important points 1 what is the minimum penalty and how does it compare to the penalty actually used in the svm algorithm 2 is it possible to obtain oracle inequalities in that setting for the specific loss function used in the svm algorithm we show that the answer to the latter question is positive and provides relevant insight to the former our result shows that it is possible to obtain fast rates of convergence for svms

isoperimetric inequalities and markov chains soit a i j i j n la matrice stochastique d une chaine de markov reversible sur n 1 2 soit n 2 tel que f r c f d f c 0 n ou r 2n n 2 ou c 0 est independant de f et ou r indique la norme dans l r n d pour la mesure i i alors on a sup i j j 1 a i j 0 n 2

random mappings of scaled graphs given a connected finite graph with a fixed base point o and some graph g with a based point we study random 1 lipschitz maps of a scaled into g we are mostly interested in the case where g is a cayley graph of some finitely generated group where the construction does not depend on the choice of base points a particular case of being a graph on two vertices and one edge corresponds to the random walk on g and the case where is a graph on two vertices and two edges joining them corresponds to brownian bridge in g we show that unlike in the case g mathbb z d the asymptotic behavior of a random scaled mapping of into g may differ significantly from the asymptotic behavior of random walks or random loops in g in particular we show that this occurs when g is a free non abelian group also we consider the case when g is a wreath product of mathbb z with a finite group to treat this case we prove new estimates for transition probabilities in such wreath products for any group g generated by a finite set s we define a functor e from category of finite connected graphs to the category of equivalence relations on such graphs given a finite connected graph the value eg s can be viewed as an asymptotic invariant of g

asymptotic behaviors of support vector machines with gaussian kernel support vector machines svms with the gaussian rbf kernel have been popular for practical use model selection in this class of svms involves two hyperparameters the penalty parameter c and the kernel width this letter analyzes the behavior of the svm classifier when these hyperparameters take very small or very large values our results help in understanding the hyperparameter space that leads to an efficient heuristic method of searching for hyperparameter values with small generalization errors the analysis also indicates that if complete model selection using the gaussian kernel has been conducted there is no need to consider linear svm

empirical margin distributions and bounding the generalization error of combined classifiers we prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers such combinations could be implemented by neural networks or by voting methods of combining the classifiers such as boosting and bagging the bounds are in terms of the empirical distribution of the margin of the combined classifier they are based on the methods of the theory of gaussian and empirical processes comparison inequalities symmetrization method concentration inequalities and they improve previous results of bartlett 1998 on bounding the generalization error of neural networks in terms of ell 1 norms of the weights of neurons and of schapire freund bartlett and lee 1998 on bounding the generalization error of boosting we also obtain rates of convergence in levy distance of empirical margin distribution to the true margin distribution uniformly over the classes of classifiers and prove the optimality of these rates

carne varopoulos bounds for centered random walks we extend the carne varopoulos upper bound on the probability transitions of a markov chain to a certain class of nonreversible processes by introducing the definition of a centering measure in the case of random walks on a group we study the connections between different notions of centering

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

infinitely many solutions for some nonlinear scalar system of two elliptic equations abstract in this paper we consider the elliptic system of two equations in h 1 r n h 1 r n u a x u 2 u 2 u v v b x v 2 u v 2 v where 1 satisfy 2 n n 2 n 3 the potentials a x b x are regular functions such that lim inf x a x a 0 and lim inf x b x b 0 moreover a x b x verify suitable decay assumptions but not requiring any symmetry property by means of the standard critical point theory we find infinitely many approximate solutions in bounded balls then we obtain infinitely many solutions for the original elliptic system by analyzing the structures of the approximate solutions carefully and by passing to a limit

asymptotically linear schrodinger equation with potential vanishing at infinity we are concerned with the existence of bound states and ground states of the following nonlinear schrodinger equation

quantile regression in reproducing kernel hilbert spaces in this article we consider quantile regression in reproducing kernel hilbert spaces which we call kernel quantile regression kqr we make three contributions 1 we propose an efficient algorithm that computes the entire solution path of the kqr with essentially the same computational cost as fitting one kqr model 2 we derive a simple formula for the effective dimension of the kqr model which allows convenient selection of the regularization parameter and 3 we develop an asymptotic theory for the kqr model

random walks driven by low moment measures we study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition for any locally compact unimodular group g and any positive function g 0 we introduce a function g which describes the fastest possible decay of n 2n e when is a symmetric continuous probability density such that is finite we estimate g for a variety of groups g and functions when is of the form with 0 0 a fixed increasing function and g 0 a natural word length measuring the distance to the identity element in g g can be thought of as a group invariant

which properties of a random sequence are dynamically sensitive consider a sequence of i i d random variables where each variable is refreshed i e replaced by an independent variable with the same law independently according to a poisson clock at any fixed time t the resulting sequence has the same law as at time 0 but there can be exceptional random times at which certain almost sure properties of the time 0 sequence are violated we prove that there are no such exceptional times for the law of large numbers and the law of the iterated logarithm so these laws are dynamically stable however there are times at which run lengths are exceptionally long that is run tests are dynamically sensitive we obtain a multifractal analysis of exceptional times for run lengths and for prediction in particular starting from an i i d sequence of unbiased random bits the random set of times t where log2 n bits among the first n bits can be predicted from their predecessors has hausdorff dimension 1 a s finally we consider simple random walk in the lattice zd and prove that transience is dynamically stable for d 5 and dynamically sensitive for d 3 4 moreover for d 3 4 the nonempty random set of exceptional times t where the walk is recurrent has hausdorff dimension 4 d 2 a s

on a class of quasilinear elliptic systems with critical growth abstract a class of quasilinear elliptic systems with critical growth are considered in a bounded domain employing critical point theory and concentration estimates we establish the existence of infinitely many solutions in higher values of dimension n p 2 p 1 p n

nonparametric quantile estimation in regression the desired estimate of y x is not always given by a conditional mean although this is most common sometimes one wants to obtain a good estimate that satisfies the property that a proportion of y x will be below the estimate for 0 5 this is an estimate of the median what might be called median regression is subsumed under the term quantile regression we present a nonparametric version of a quantile estimator which can be obtained by solving a simple quadratic programming problem and provide uniform convergence statements and bounds on the quantile property of our estimator experimental results show the feasibility of the approach and competitiveness of our method with existing ones we discuss several types of extensions including an approach to solve the quantile crossing problems as well as a method to incorporate prior qualitative knowledge such as monotonicity constraints

minorations pour les chaines de markov unidimensionnelles we prove a space time lower estimate for reversible random walks on graphs having linear growth this estimate complements a similar upper bound that follows from a work of hebisch and saloff coste

stability and generalization we define notions of stability for learning algorithms and show how to use these notions to derive generalization error bounds based on the empirical error and the leave one out error the methods we use can be applied in the regression framework as well as in the classification one when the classifier is obtained by thresholding a real valued function we study the stability properties of large classes of learning algorithms such as regularization based algorithms in particular we focus on hilbert space regularization and kullback leibler regularization we demonstrate how to apply the results to svm for regression and classification

a dirichlet principle for non reversible markov chains and some recurrence theorems we extend the dirichlet principle to non reversible markov processes on countable state spaces we present two variational formulas for the solution of the poisson equation or equivalently for the capacity between two disjoint sets as an application we prove a some recurrence theorems in particular we show the recurrence of two dimensional cycle random walks under a second moment condition on the winding numbers

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

on a class of asymptotically linear elliptic problems in rn under these conditions p is an asymptotically linear problem when this equation is considered in a bounded domain 0 r with say the dirichlet boundary condition there is a large literature on existence and multiplicity results of particular interest is then the case of resonance where s and s is the asymptotic linearization of the problem in other words s d s l 0 l 0 is the operator given by

characterization of sub gaussian heat kernel estimates on strongly recurrent graphs sub gaussian estimates for random walks are typical of fractal graphs we characterize them in the strongly recurrent case in terms of resistance estimates only without assuming elliptic harnack inequalities

small time gaussian estimates of heat diffusion kernels ii the theory of large deviations la deuxieme partie d un article etudiant le noyau de chaleur d un operateur differentiel d ordre 2 sur une variete c

hr bounds for spectral multipliers on graphs we study the boundedness on the hardy spaces h p of spectral multiplier operators associated with the discrete laplacian on a weighted graph we assume that the graph satisfies the doubling volume property and a poincare inequality we prove that there is p 0 0 1 depending on the geometry of the graph such that if the multiplier satisfies a condition similar to the one we have in the classical hormander multiplier theorem then the corresponding operator is bounded on h p p p 0 1

markov chain approximations to nonsymmetric diffusions with bounded coefficients we consider a certain class of nonsymmetric markov chains and obtain heat kernel bounds and parabolic harnack inequalities using the heat kernel estimates we establish a sufficient condition for the family of markov chains to converge to nonsymmetric diffusions as an application we approximate nonsymmetric diffusions in divergence form with bounded coefficients by nonsymmetric markov chains this extends the results by stroock and zheng to the nonsymmetric divergence forms 2012 wiley periodicals inc

tree indexed random walks on groups and first passage percolation suppose that i i d random variables are attached to the edges of an infinite tree when the tree is large enough the partial sumss along some of its infinite paths will exhibit behavior atypical for an ordinary random walk this principle has appeared in works on branching random walks first passage percolation and rwre on trees we establish further quantitative versions of this principle which are applicable in these settings in particular different notions of speed for such a tree indexed walk correspond to different dimension notions for trees finally if the labeling variables take values in a group then properties of the group e g polynomial growth or a nontrivial poisson boundary are reflected in the sample path behavior of the resulting tree indexed walk

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

the existence of a positive solution to asymptotically linear scalar field equations which is important in using the mountain pass theorem the aim of this paper is to discuss how to use the mountain pass theorem to show the existence of a positive solution to the present problem when we lose the above condition furthermore if f x u f u we also prove that the above problem has a ground state by using the articial constraint method

analyticity and discrete maximal regularity on lp spaces abstract in the first part of this paper we give the following interpolation result on the analyticity i e the property t i tn c n for all n n of an operator t on lp if t is powerbounded on lp and lq as well as analytic on lp then t is powerbounded and analytic on lr for all r strictly between p and q this is a discrete analogue of the well known corresponding result for analytic semigroups eta as recently shown by the author the analyticity of t is a necessary condition for the maximal regularity of the discrete time evolution equation un 1 tun fn for all n z u0 0 in the second part of this paper we establish the following two sufficient conditions for its maximal regularity t is a subpositive analytic contraction or t is an integral operator satisfying certain poisson bounds these results are discrete analogues of the corresponding results for the maximal regularity of the evolution equation u t au t f t for all t r u 0 0 due to lamberton weis coulhon and duong and hieber and pruss for the poisson bound result of coulhon and duong and hieber and pruss we give a slight improvement and a short proof

indistinguishability of trees in uniform spanning forests we prove that in both the free and the wired uniform spanning forest fusf and wusf of any unimodular random rooted network in particular of any cayley graph it is impossible to distinguish the connected components of the forest from each other by invariantly defined graph properties almost surely this confirms a conjecture of benjamini et al ann probab 29 1 1 65 2001 we also answer positively two additional questions of benjamini et al ann probab 29 1 1 65 2001 under the assumption of unimodularity we prove that on any unimodular random rooted network the fusf is either connected or has infinitely many connected components almost surely and if the fusf and wusf are distinct then every component of the fusf is transient and infinitely ended almost surely all of these results are new even for cayley graphs

the local limit theorem on nilpotent lie groups a local limit theorem is proven on connected simply connected nilpotent lie groups for a class of generating measures satisfying a moment condition and a condition on the characteristic function of the abelianization the result extends an earlier local limit theorem of alexopoulos which treated absolutely continuous measures with a continuous density of compact support and also extends local limit theorems of breuillard and diaconis hough which treated general measures on the heisenberg group

local rademacher complexities and oracle inequalities in risk minimization let be a class of measurable functions f s 0 1 defined on a probability space s mathcal a p given a sample x1 xn of i i d random variables taking values in s with common distribution p let pn denote the empirical measure based on x1 xn we study an empirical risk minimization problem pnf min f given a solution fn of this problem the goal is to obtain very general upper bounds on its excess risk mathcal e p hat f n p hat f n inf f in mathcal f pf expressed in terms of relevant geometric parameters of the class using concentration inequalities and other empirical processes tools we obtain both distribution dependent and data dependent upper bounds on the excess risk that are of asymptotically correct order in many examples the bounds involve localized sup norms of empirical and rademacher processes indexed by functions from the class we use these bounds to develop model selection techniques in abstract risk minimization problems that can be applied to more specialized frameworks of regression and classification

centered densities on lie groups of polynomial volume growth we study the asymptotic behavior of the convolution powers of a centered density on a connected lie group g of polynomial volume growth the main tool is a harnack inequality which is proved by using ideas from homogenization theory and by adapting the method of krylov and safonov applying this inequality we prove that the positive harmonic functions are constant we also characterise the harmonic functions which grow polynomially we give gaussian estimates for as well as for the differences and we give estimates similar to the ones given by the classical berry esseen theorem for and we use these estimates to study the associated riesz transforms

analysis on nilpotent groups soit g un groupe de lie nilpotent reel simplement connexe et soit l son algebre de lie soient l 1 l et l k 1 l k l k 1 2 soit d d g d l k 1 kdim r l k l k 1 d est la dimension de dirichlet de g soit h un groupe nilpotent discret finiment genere on demontre des theoremes d analyse pour g

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

an explicit description of the reproducing kernel hilbert spaces of gaussian rbf kernels although gaussian radial basis function rbf kernels are one of the most often used kernels in modern machine learning methods such as support vector machines svms little is known about the structure of their reproducing kernel hilbert spaces rkhss in this work two distinct explicit descriptions of the rkhss corresponding to gaussian rbf kernels are given and some consequences are discussed furthermore an orthonormal basis for these spaces is presented finally it is discussed how the results can be used for analyzing the learning performance of svms

local asymptotics for regression splines and confidence regions in this paper we study the local behavior of regression splines in particular explicit expressions for the asymptotic pointwise bias and variance of regression splines are obtained in addition asymptotic normality for regression splines is established leading to the construction of approximate confidence intervals and confidence bands for the regression function

asymptotic entropy and green speed for random walks on countable groups we study asymptotic properties of the green metric associated with transient random walks on countable groups we prove that the rate of escape of the random walk computed in the green metric equals its asymptotic entropy the proof relies on integral representations of both quantities with the extended martin kernel in the case of finitely generated groups where this result is known benjamini and peres probab theory related fields 98 1994 91 112 we give an alternative proof relying on a version of the so called fundamental inequality relating the rate of escape the entropy and the logarithmic volume growth extended to random walks with unbounded support

free knot spline smoothing for functional data summary the paper introduces free knot regression spline estimators for the mean and the variance components of a sample of curves the asymptotic distribution of the mean estimator is derived and asymptotic confidence bands are constructed a comparative simulation study shows that free knot splines estimate salient features of the functions such as sharp peaks more accurately than smoothing splines this adaptive behaviour is also illustrated by an analysis of weather data

positive solutions for a nonhomogeneous elliptic equation on rn without ar condition abstract in this paper we study the following problem 0 1 u u k x f u h x x r n u h 1 r n u 0 in r n n 3 where f t is either asymptotically linear or superlinear with respect to t at infinity the ambrosetti rabinowitz type condition that is so called ar condition ar 0 f t 0 t f s d s t f t for t 0 and some 0 1 2 as well as the monotonicity of f t t are not assumed under appropriate assumptions on k h and f we prove that problem 0 1 has at least two positive solutions

on l1 norm multiclass support vector machines binary support vector machines svms have been proven to deliver high performance in multiclass classification however issues remain with respect to variable selection one challenging issue is classification and variable selection in the presence of variables in the magnitude of thousands greatly exceeding the size of training sample this often occurs in genomics classification to meet the challenge this article proposes a novel multiclass support vector machine which performs classification and variable selection simultaneously through an l1 norm penalized sparse representation the proposed methodology together with the developed regularization solution path permits variable selection in such a situation for the proposed methodology a statistical learning theory is developed to quantify the generalization error in an attempt to gain insight into the basic structure of sparse learning permitting the number of variables to greatly exceed the sample size the operating characteristics of the

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

rate of escape of random walks on wreath products and related groups this article examines the rate of escape for a random walk on g z and proves laws of the iterated logarithm for both the inner and outer radius of escape the class of g for which these results hold includes finite g as well as groups of the form h z so the construction can be iterated laws of the iterated logarithm are also found for random walk on baumslag solitar groups and a discrete version of the sol geometry

dimension free lp estimates for single riesz transforms via an h joint functional calculus abstract by using an h joint functional calculus for strongly commuting operators we derive a scheme to deduce the l p boundedness of certain d dimensional riesz transforms from the l p boundedness of appropriate one dimensional riesz transforms moreover the l p bounds we obtain are independent of the dimension the scheme is applied to riesz transforms connected with orthogonal expansions and discrete riesz transforms on products of groups with polynomial growth

conditional growth charts growth charts are often more informative when they are customized per subject taking into account prior measurements and possibly other covariates of the subject we study a global semiparametric quantile regression model that has the ability to estimate conditional quantiles without the usual distributional assumptions the model can be estimated from longitudinal reference data with irregular measurement times and with some level of robustness against outliers and it is also flexible for including covariate information we propose a rank score test for large sample inference on covariates and develop a new model assessment tool for longitudinal growth data our research indicates that the global model has the potential to be a very useful tool in conditional growth chart analysis

local rademacher complexities we propose new bounds on the error of learning algorithms in terms of a data dependent notion of complexity the estimates we establish give optimal rates and are based on a local and empirical version of rademacher averages in the sense that the rademacher averages are computed from the data on a subset of functions with small empirical error we present some applications to classification and prediction with convex function classes and with kernel classes in particular

bivariate quantile smoothing splines it has long been recognized that the mean provides an inadequate summary whereas the set of quantiles can supply a more complete description of a sample we introduce bivariate quantile smoothing splines which belong to the space of bilinear tensor product splines as nonparametric estimators for the conditional quantile functions in a two dimensional design space the estimators can be computed by using standard linear programming techniques and can further be used as building blocks for conditional quantile estimations in higher dimensions for moderately large data sets we recommend penalized bivariate b splines as approximate solutions we use real and simulated data to illustrate the methodology proposed

the component graph of the uniform spanning forest transitions in dimensions 9 10 11 ldots 9 10 11 we prove that the uniform spanning forests of mathbb z d and mathbb z ell have qualitatively different connectivity properties whenever ell d ge 4 in particular we consider the graph formed by contracting each tree of the uniform spanning forest down to a single vertex which we call the component graph we introduce the notion of ubiquitous subgraphs and show that the set of ubiquitous subgraphs of the component graph changes whenever the dimension changes and is above 8 to separate dimensions 5 6 7 and 8 we prove a similar result concerning ubiquitous subhypergraphs in the component hypergraph our result sharpens a theorem of benjamini kesten peres and schramm who proved that the diameter of the component graph increases by one every time the dimension increases by four

quantile smoothing splines summary although nonparametric regression has traditionally focused on the estimation of conditional mean functions nonparametric estimation of conditional quantile functions is often of substantial practical interest we explore a class of quantile smoothing splines defined as solutions to

detecting differential expressions in genechip microarray studies a quantile approach in this article we consider testing for differentially expressed genes in genechip studies by modeling and analyzing the quantiles of gene expression through probe level measurements by developing a robust rank score test for linear quantile models with a random effect we propose a reliable test for detecting differences in certain quantiles of the intensity distributions by using a genomewide adjustment to the test statistic to account for within array correlation we demonstrate that the proposed rank score test is highly effective even when the number of arrays is small our empirical studies with real experimental data show that detecting differences in the quartiles for the probe level data is a valuable complement to the usual mixed model analysis based on gaussian likelihood the methodology proposed in this article is a first attempt to develop inferential tools for quantile regression in mixed models

on quantile regression in reproducing kernel hilbert spaces with the data sparsity constraint for spline regressions it is well known that the choice of knots is crucial for the performance of the estimator as a general learning framework covering the smoothing splines learning in a reproducing kernel hilbert space rkhs has a similar issue however the selection of training data points for kernel functions in the rkhs representation has not been carefully studied in the literature in this paper we study quantile regression as an example of learning in a rkhs in this case the regular squared norm penalty does not perform training data selection we propose a data sparsity constraint that imposes thresholding on the kernel function coefficients to achieve a sparse kernel function representation we demonstrate that the proposed data sparsity method can have competitive prediction performance for certain situations and have comparable performance in other cases compared to that of the traditional squared norm penalty therefore the data sparsity method can serve as a competitive alternative to the squared norm penalty method some theoretical properties of our proposed method using the data sparsity constraint are obtained both simulated and real data sets are used to demonstrate the usefulness of our data sparsity constraint