asymptotic equivalence of nonparametric regression and white noise the principal result is that under conditions to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems and conversely this asymptotic equivalence is in a global and uniform sense any normalized risk function attainable in one problem is asymptotically attainable in the other with the difference in normalized risks converging to zero uniformly over the entire parameter space the results are constructive a recipe is provided for producing these asymptotically equivalent procedures some implications and generalizations of the principal result are also discussed

phase transitions for high dimensional clustering and related problems consider a two class clustering problem where we observe x i ell i mu z i z i stackrel mathit i i d sim n 0 i p 1 leq i leq n the feature vector mu in r p is unknown but is presumably sparse the class labels ell i in 1 1 are also unknown and the main interest is to estimate them we are interested in the statistical limits in the two dimensional phase space calibrating the rarity and strengths of useful features we find the precise demarcation for the region of impossibility and region of possibility in the former useful features are too rare weak for successful clustering in the latter useful features are strong enough to allow successful clustering the results are extended to the case of colored noise using le cam s idea on comparison of experiments we also extend the study on statistical limits for clustering to that for signal recovery and that for global testing we compare the statistical limits for three problems and expose some interesting insight we propose classical pca and important features pca if pca for clustering for a threshold t 0 if pca clusters by applying classical pca to all columns of x with an l 2 norm larger than t we also propose two aggregation methods for any parameter in the region of possibility some of these methods yield successful clustering we discover a phase transition for if pca for any threshold t 0 let xi t be the first left singular vector of the post selection data matrix the phase space partitions into two different regions in one region there is a t such that cos xi t ell rightarrow 1 and if pca yields successful clustering in the other cos xi t ell leq c 0 0 our results require delicate analysis especially on post selection random matrix theory and on lower bound arguments

bayesian aspects of some nonparametric problems we study the bayesian approach to nonparametric function estimation problems such as nonparametric regression and signal estimation we consider the asymptotic properties of bayes procedures for conjugate gaussian priors we show that so long as the prior puts nonzero measure on the very large parameter set of interest then the bayes estimators are not satisfactory more specifically we show that these estimators do not achieve the correct minimax rate over norm bounded sets in the parameter space thus all bayes estimators for proper gaussian priors have zero asymptotic efficiency in this minimax sense we then present a class of priors whose bayes procedures attain the optimal minimax rate of convergence these priors may be viewed as compound or hierarchical mixtures of suitable gaussian distributions

projection estimation in multiple regression with application to functional anova models a general theory on rates of convergence of the least squares projection estimate in multiple regression is developed the theory is applied to the functional anova model where the multivariate regression function is modeled as a specified sum of a constant term main effects functions of one variable and selected interaction terms functions of two or more variables the least squares projection is onto an approximating space constructed from arbitrary linear spaces of functions and their tensor products respecting the assumed anova structure of the regression function the linear spaces that serve as building blocks can be any of the ones commonly used in practice polynomials trigonometric polynomials splines wavelets and finite elements the rate of convergence result that is obtained reinforces the intuition that low order anova modeling can achieve dimension reduction and thus overcome the curse of dimensionality moreover the components of the projection estimate in an appropriately defined anova decomposition provide consistent estimates of the corresponding components of the regression function when the regression function does not satisfy the assumed anova form the projection estimate converges to its best approximation of that form

variable bandwidth kernel estimators of regression curves on montre qu en termes de l erreur en moyenne quadratique integree asymptotique les estimateurs du noyau avec un certain choix de largeur de bande local sont superieurs aux estimateurs ordinaires du noyau avec un choix de largeur de bande global si on utilise des largeurs de bandes optimales

conjugacy growth of commutators abstract for the free group f r on r 1 generators respectively the free product g 1 g 2 of two nontrivial finite groups g 1 and g 2 we obtain the asymptotic for the number of conjugacy classes of commutators in f r respectively g 1 g 2 with a given word length in a fixed set of free generators respectively the set of generators given by the nontrivial elements of g 1 and g 2 our result is proven by using the classification of commutators in free groups and in free products by wicks and builds on the works of rivin and sharp who asymptotically counted the conjugacy classes of commutator subgroup elements in f r with a given word length

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

regularization of wavelet approximations in this paper we introduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced the approach can apply readily to many other statistical contexts various new penalty functions are proposed the hard thresholding and soft thresholding estimators of donoho and johnstone are specific members of nonlinear regularized wavelet estimators they correspond to the lower and upper envelopes of a class of the penalized least squares estimators necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions the sampling properties of nonlinear regularized wavelet estimators are established and are shown to be adaptively minimax to efficiently solve penalized least squares problems nonlinear regularized sobolev interpolators nrsi are proposed as initial estimators wh

on the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

asymptotic equivalence theory for nonparametric regression with random design this paper establishes the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function the asymptotic equivalence is established by constructing explicit equivalence mappings between the nonparametric regression and the whitenoise experiments which provide synthetic observations and synthetic asymptotic solutions from any one of the two experiments with asymptotic properties identical to the true observations and given asymptotic solutions from the other the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in all other asymptotically equivalent nonparametric problems 1 introduction the purpose of this paper is to establish the global asymptotic equivalence between the nonparametric regression with random design and the white noise under sharp smoothness conditions on an unknown regression or drift function we establish this asymptotic equivalence by constructing explicit equivalence mappings between the nonparametric regression and the white noise problems as in brown and low 1996 for their asymptotic equivalence results the equivalence mapping from the nonparametric regression to the white noise provides synthetic observations of the white noise from the nonparametric regression such that the distributions of the synthetic observations are asymptotically equivalent to those of the true observations of the white noise for any asymptotic solution to a white noise problem the application of the solution to the synthetic observations provides an asymptotic solution to the corresponding nonparametric regression problem with identical asymptotic properties likewise the equivalence mapping from the white noise produces synthetic observations of the nonparametric regression problem and synthetic asymptotic solutions to white noise problems based on those of the corresponding nonparametric regression problems the impact of such asymptotic equivalence results is that an investigation in one nonparametric problem automatically yields asymptotically analogous results in

the growth of free products abstract we improve estimates for the exponential growth rates of free products with or without amalgamation hnn extensions one relator groups and groups of positive deficiency some examples of groups for which the growth rates are realized are given

a discrete transform and decompositions of distribution spaces abstract we study a representation formula of the form q q q for a distribution on r n this formula is obtained by discretizing and localizing a standard littlewood paley decomposition the map taking to the sequence q q with q running over the dyadic cubes in r n is called the transform the functions q and q have a particularly simple form moreover most of the familiar distribution spaces lp spaces 1 r n we also consider pointwise multipliers for the characteristic function of a domain we obtain boundedness results for a general class of domains which properly includes lipschitz domains several interpolation methods are easily analyzed via the sequence spaces for real interpolation we obtain among other things an extension to the case p 0 this in turn gives a new approach to the traditional atomic decomposition of hardy spaces

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

minimax risk over hyperrectangles and implications consider estimating the mean of a standard gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in l 2 such sets include ellipsoids hyperrectangles and l p bodies with p 2 the minimax risk among linear estimates is within 25 of the minimax risk among all estimates the minimax risk among truncated series estimates is within a factor 4 44 of the minimax risk this implies that the difficulty of estimation a statistical quantity is measured fairly precisely by the n widths a geometric quantity if the set is not quadratically convex as in the case of l p bodies with p 2 things change appreciably minimax linear estimators may be out performed arbitrarily by nonlinear estimates the ordinary kolmogorov n widths still determine the difficulty of linear estimation but the difficulty of nonlinear estimation is tied to the inner bernstein n widths which can be far smaller essential use is made of a new heuristic that the difficulty of the hardest rectangular subproblem is equal to the difficulty of the full problem

adaptive wavelet estimation a block thresholding and oracle inequality approach we study wavelet function estimation via the approach of block thresh olding and ideal adaptation with oracle oracle inequalities are derived and serve as guides for the selection of smoothing parameters based on an oracle inequality and motivated by the data compression and localization properties of wavelets an adaptive wavelet estimator for nonparametric re gression is proposed and the optimality of the procedure is investigated we show that the estimator achieves simultaneously three objectives adaptiv ity spatial adaptivity and computational efficiency specifically it is proved that the estimator attains the exact optimal rates of convergence over a range of besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point the estimator is easy to imple ment at the computational cost of on simulation shows that the es timator has excellent numerical performance relative to more traditional wavelet estimators 1 introduction wavelet methods have demonstrated considerable suc cess in nonparametric function estimation in terms of spatial adaptivity com putational efficiency and asymptotic optimality in contrast to the traditional linear procedures wavelet methods achieve near optimal convergence rates over large function classes such as besov classes and enjoy excellent mean squared error properties when used to estimate functions that are spatially inhomogeneous for example as shown by donoho and johnstone 1998 wavelet methods can outperform optimal linear methods even at the level of convergence rate over certain besov classes standard wavelet methods achieve adaptivity through term by term thresholding of the empirical wavelet coefficients there each individual empirical wavelet coefficient is compared with a predetermined threshold a wavelet coefficient is retained if its magnitude is above the threshold level and is discarded otherwise a well known example of term by term thresholding is donoho and johnstone s visushrink donoho and johnstone 1994 visushrink is spatially adaptive and the estimator is within a log arithmic factor of the optimal convergence rate over a wide range of besov classes visushrink achieves a degree of tradeoff between variance and bias contributions to the mean squared error however the tradeoff is not optimal visushrink reconstruction is often over smoothed hall kerkyacharian and picard 1999 considered block thresholding for wavelet function estimation which thresholds empirical wavelet coefficients in

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

on the choice of smoothing parameter threshold and truncation in nonparametric regression by non linear wavelet methods summary concise asymptotic theory is developed for non linear wavelet estimators of regression means in the context of general error distributions general designs general normalizations in the case of stochastic design and non structural assumptions about the mean the influence of the tail weight of the error distribution is addressed in the setting of choosing threshold and truncation parameters mainly the tail weight is described in an extremely simple way by a moment condition previous work on this topic has generally imposed the much more stringent assumption that the error distribution be normal different approaches to correction for stochastic design are suggested these include conventional kernel estimation of the design density in which case the interaction between the smoothing parameters of the non linear wavelet estimator and the linear kernel method is described

relative growth series in some hyperbolic groups we study certain groups of isometries of hyperbolic space including certain hyperbolic coxeter groups and normal subgroups with quotient z 1 we show that the associated relative growth series are not rational this extends results obtained by grigorchuk and de la harpe for free groups and pollicott and the author for surface groups

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

minimizing the total variation under a general convex constraint for image restoration we present a general framework for image restoration despite its simplicity certain variational and certain wavelet approaches can be formulated within this framework this permits the construction of a natural model with only one parameter which has the advantages of both approaches we give a mathematical analysis of this model describe our algorithm and illustrate this by some experiments

denoising methods for thermomechanical decomposition for quasi equilibrium molecular dynamics simulations in this work we applied robust denoising methods well established in the signal processing field for the thermomechanical decomposition of velocity data obtained from molecular dynamics md simulations in the decomposition the atomic velocity was assumed to be the sum of the mechanical velocity and the thermal velocity which can be linked to the stress and temperature field at the continuum scale respectively for the quasi equilibrium process with the thermal velocity treated as the gaussian distributed stationary noise with zero mean and a positive variance that is linearly proportional to temperature and mechanical velocity as the clean signal the velocity decomposition can be recasted into a denoising problem for which powerful denoising methods have been developed to estimate a clean signal from noisy data we investigated the widely used linear parametric real domain linear nonparametric fourier based and nonlinear nonparametric wavelet based denoising methods first on their theoretical properties and then made comparsion among them for denosing some synthetic noisy 1 dimensional 1d data generated from md simulations the nonlinear wavelet based thresholding estimator possessed better optimality properties than the other estimators and also outperformed the other estimators in the synthetic data test a further test comparing the various denoising methods for an adiabatic shear crack nucleation and propagation process simulated using md simulations showed better performance by the wavelet based denoisng method results from this work reveal good potential of applying wavelet based denoising method to the study of thermomechanical processes simulated using md simulations

stable commutator length is rational in free groups for any group there is a natural pseudo norm on the vector space b h 1 of real homogenized group 1 boundaries called the stable commutator length norm this norm is closely related to and can be thought of as a relative version of the gromov pseudo norm on ordinary homology we show that for a free group the unit ball of this pseudo norm is a rational polyhedron it follows that the stable commutator length in free groups takes on only rational values moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group the proof of these facts yields an algorithm to compute the stable commutator length in free groups using this algorithm we answer a well known question of bavard in the negative constructing explicit examples of elements in free groups whose stable commutator length is not a half integer

local limit theorems for free groups in this paper we obtain a local limit theorem for elements of a free group g under the abelianization map cdot g to g g g this is obtained via an analysis involving subshifts of finite type where we obtain a result of independent interest the case of fundamental groups of compact surfaces of genus geq 2 is also discussed

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed