component selection and smoothing in multivariate nonparametric regression we propose a new method for model selection and model fitting in multivariate nonparametric regression models in the framework of smoothing spline anova the cosso is a method of regularization with the penalty functional being the sum of component norms instead of the squared norm employed in the traditional smoothing spline method the cosso provides a unified framework for several recent proposals for model selection in linear models and smoothing spline anova models theoretical properties such as the existence and the rate of convergence of the cosso estimator are studied in the special case of a tensor product design with periodic functions a detailed analysis reveals that the cosso does model selection by applying a novel soft thresholding type operation to the function components we give an equivalent formulation of the cosso estimator which leads naturally to an iterative algorithm we compare the cosso with mars a popular method that builds functional anova models in simulations and real examples the cosso method can be extended to classification problems and we compare its performance with those of a number of machine learning algorithms on real datasets the cosso gives very competitive performance in these studies

multiplicity results for asymptotically linear elliptic problems at resonance in this paper several new multiplicity results for asymptotically linear elliptic problem at resonance are obtained via morse theory and minimax methods some new observations on the critical groups of a local linking type critical point are used to deal with the resonance case at 0

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

sparse additive models summary we present a new class of methods for high dimensional non parametric regression and classification called sparse additive models our methods combine ideas from sparse linear modelling and additive non parametric regression we derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size sparse additive models are essentially a functional version of the grouped lasso of yuan and lin they are also closely related to the cosso model of lin and zhang but decouple smoothing and sparsity enabling the use of arbitrary non parametric smoothers we give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data showing that they can be effective in fitting sparse non parametric models in high dimensional data

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

multiple nontrivial solutions of a semilinear elliptic problem with asymmetric nonlinearity abstract in this paper we study the existence and multiplicity of solutions of the boundary value problem 1 u u q 2 u a u b u p 1 in u 0 on where denotes the n dimensional laplacian is a bounded domain with smooth boundary in r n n 3 u denotes the positive part of u r 1 q 2 p 2 2 n n 2 0 a r and b 0 using infinite dimensional morse theory we extend the results of paiva and presoto 14 and establish some conditions for the existence of at least four nontrivial solutions of 1

multiple periodic solutions of ordinary differential equations with double resonance abstract in this paper we study by morse theory the existence of multiple periodic solutions of a class of ordinary differential equation with double resonance at infinity between two consecutive eigenvalues and with resonance at origin

dirichlet problems with double resonance and an indefinite potential abstract we consider semilinear dirichlet problems with an unbounded and indefinite potential and with a caratheodory reaction we assume that asymptotically at infinity the problem exhibits double resonance using variational methods together with morse theory and flow invariance arguments we prove multiplicity theorems producing three five six or seven nontrivial smooth solutions in most multiplicity theorems we provide precise sign information for all the solutions established

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

critical groups at zero and multiple solutions for a quasilinear elliptic equation abstract in this paper by morse theory we will compute the critical groups at zero for a functional i w 0 1 p r defined by setting i u 1 p u p d x 1 2 u 2 d x f x u d x where p 2 is a bounded domain in r n f x u 0 u f x t d t and we assume that f is resonant at zero for the spectrum of in w 0 1 2 as an application of these critical groups estimates some multiplicity results are also given

an elliptic resonance problem with multiple solutions in this paper we study the existence of multiple nontrivial solutions for a semilinear elliptic equation at resonance by the minimax methods and morse theory

multiple solutions for elliptic problems with asymmetric nonlinearity in this paper we establish the existence of multiple solutions for the semilinear elliptic problem u g x u in u 0on where rn is an open bounded domain with smooth boundary g r r is a function of class c1 such that g x 0 0 and which is asymptotically linear at infinity with jumping nonlinearities we use critical groups to distinguish the critical points

asymptotically linear elliptic equations without nonresonance conditions abstract in this paper we are concerned with the existence of nontrivial solutions of a semilinear elliptic equation when the nonresonance condition at infinity is not satisfied we consider a strongly resonant case in the first part and a sort of resonance under density assumptions in the second part

pairs of nontrivial solutions for resonant neumann problems abstract we study a semilinear neumann problem which is resonant at with respect to any eigenvalue different from the first and the second eigenvalue of n the negative neumann laplacian using a combination of variational methods with morse theoretic techniques we show that the problem has at least two nontrivial smooth solutions

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

estimation of a sparse group of sparse vectors summary we consider estimating a sparse group of sparse normal mean vectors based on penalized likelihood estimation with complexity penalties on the number of nonzero mean vectors and the numbers of their significant components which can be performed by a fast algorithm the resulting estimators are developed within a bayesian framework and can be viewed as maximum a posteriori estimators we establish their adaptive minimaxity over a wide range of sparse and dense settings a simulation study demonstrates the efficiency of the proposed approach which successfully competes with the sparse group lasso estimator

multiple solutions for semilinear resonance elliptic problems with nonconstant coefficients at infinity in this paper we study a class of semilinear elliptic equations which are completely resonant with nonconstant coefficients at origin and at infinity by applying morse theory and minimax method we obtain the existence of multiple nontrivial solutions for the equations

sparse additive regression on a regular lattice summary we consider estimation in a sparse additive regression model with the design points on a regular lattice we establish the minimax convergence rates over sobolev classes and propose a fourier based rate optimal estimator which is adaptive to the unknown sparsity and smoothness of the response function the estimator is derived within a bayesian formalism but can be naturally viewed as a penalized maximum likelihood estimator with the complexity penalties on the number of non zero univariate additive components of the response and on the numbers of the non zero coefficients of their fourer expansions we compare it with several existing counterparts and perform a short simulation study to demonstrate its performance

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

multiple solutions for semilinear elliptic boundary value problems with double resonance abstract in this paper we study the multiplicity of nontrivial solutions of semilinear elliptic boundary value problems which may be double resonance near infinity between two consecutive eigenvalues of with zero dirichlet boundary data the methods we use here are morse theory minimax methods and bifurcation theory

on optimality of bayesian testimation in the normal means problem we consider a problem of recovering a high dimensional vector observed in white noise where the unknown vector is assumed to be sparse the objective of the paper is to develop a bayesian formalism which gives rise to a family of l 0 type penalties the penalties are associated with various choices of the prior distributions n on the number of nonzero entries of and hence are easy to interpret the resulting bayesian estimators lead to a general thresholding rule which accommodates many of the known thresholding and model selection procedures as particular cases corresponding to specific choices of n furthermore they achieve optimality in a rather general setting under very mild conditions on the prior we also specify the class of priors n for which the resulting estimator is adaptively optimal in the minimax sense for a wide range of sparse sequences and consider several examples of such priors

on bayesian testimation and its application to wavelet thresholding summary we consider the problem of estimating the unknown response function in the gaussian white noise model we first utilize the recently developed bayesian maximum a posteriori testimation procedure of abramovich et al 2007 for recovering an unknown high dimensional gaussian mean vector the existing results for its upper error bounds over various sparse lp balls are extended to more general cases we show that for a properly chosen prior on the number of nonzero entries of the mean vector the corresponding adaptive estimator is asymptotically minimax in a wide range of sparse and dense lp balls the proposed procedure is then applied in a wavelet context to derive adaptive global and level wise wavelet estimators of the unknown response function in the gaussian white noise model these estimators are then proven to be respectively asymptotically near minimax and minimax in a wide range of besov balls these results are also extended to the estimation of derivatives of the response function simulated examples are conducted to illustrate the performance of the proposed level wise wavelet estimator in finite sample situations and to compare it with several existing counterparts

semilinear elliptic problems with asymmetric nonlinearities abstract in this paper we are concerned on the semilinear elliptic problem u u q 2 u a u b u p 1 in u 0 on where r n is a bounded domain with regular boundary 1 q 2 p 2 if a is between two eigenvalues we get the existence of three nontrivial solutions for the problem above

an asymmetric superlinear elliptic problem at resonance abstract in this article we study the existence and multiplicity of solutions to the problem u g x u in u 0 on where is a bounded domain in r n n 2 with smooth boundary and g r r is a differentiable function we will assume that g x s has a resonant behavior for large negative values of s and that a landesman lazer type condition is satisfied we also assume that g x s is superlinear but subcritical for large positive values of s we prove the existence and multiplicity of solutions for problem 1 1 by using minimax methods and infinite dimensional morse theory

applications of local linking to critical point theory we prove the existence of nontrivial critical points of functionals with a local linking at 0 applications to elliptic boundary value problems and to hamiltonian systems are given c 1995 academic press inc

existence of a nontrivial solution for a class of hemivariational inequality problems at double resonance abstract in this paper we discuss a class of semilinear elliptic hemivariational inequality problems by using the nonsmooth minimax principle for locally lipschitz functions we establish the existence of a nontrivial solution for the semilinear elliptic hemivariational inequality problem where incomplete double resonance occurs at infinity between two distinct consecutive eigenvalues

on a robin problem with indefinite weight and asymmetric reaction superlinear at abstract the existence of at least three nontrivial smooth solutions to a semilinear robin problem with indefinite potential and asymmetric nonlinearity super linear at is established both crossing and resonance are allowed proofs exploit variational methods truncation techniques and morse theory

multiple solutions for boundary value problems of second order difference equations with resonance in this paper we study the existence of multiple solutions for boundary value problems of second order difference equations with resonance at both infinity and zero by using morse theory critical point theory minimax methods and bifurcation theory

model selection and sharp asymptotic minimaxity we obtain sharp minimax results for estimation of an n dimensional normal mean under quadratic loss the estimators are chosen by penalized least squares with a penalty that grows like ck log n k for k equal to the number of nonzero elements in the estimating vector for a wide range of sparse parameter spaces we show that the penalized estimator achieves the exact minimax rate with the correct multiplication constant if and only if c equals 2 our results unify the theory obtained by many other authors for penalized estimation of normal means in particular we establish that a conjecture by abramovich et al ann stat 34 584 653 2006 is true

nonlinear elliptic equations with a jumping reaction abstract we study a nonlinear nonhomogeneous dirichlet problem driven by the sum of a p laplacian and a laplacian 2 p and a jumping nonlinearity under very general conditions on the reaction and without using the fucik spectrum we show that the problem has at least three nontrivial solutions and we provide sign information for all of them our approach uses critical point theory truncation and comparison techniques and morse theory

multiple solutions to a robin problem with indefinite weight and asymmetric reaction abstract the existence of two nontrivial smooth solutions to a semilinear robin problem with indefinite unbounded potential and asymmetric nonlinearity f is established both crossing and resonance are allowed a third nonzero solution exists provided f is c 1 proofs exploit variational methods truncation techniques and morse theory

robin problems with a general potential and a superlinear reaction abstract we consider semilinear robin problems driven by the negative laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the ambrosetti rabinowitz condition we prove existence and multiplicity theorems producing also an infinity of smooth solutions using variational tools truncation and perturbation techniques and morse theory critical groups

on the computation of power generating system reliability indexes reliability models play an important role in the determination of the required generation reserves of a given electric power utility company and of its expected operating cost specifically these models are used to compute the risk of system load loss and the capacity utilization ratios of different units comprising the system computation of these indexes usually requires the evaluation of the distribution function of the sum of a large number of discrete random variables with different distributions exact methods for computing these distributions prove computationally unfeasible for routine utility operations this article compares the accuracy of several analytical approximations for computing the required performance indexes these approximations are a the method of cumulants b a distribution fitting procedure and c a large deviation procedure due to esscher application of these procedures to computing the indexes for several representative utility systems shows that esscher s large deviati

portfolio value at risk based on independent component analysis risk management technology applied to high dimensional portfolios needs simple and fast methods for calculation of value at risk var the multivariate normal framework provides a simple off the shelf methodology but lacks the heavy tailed distributional properties that are observed in data a principle component based method tied closely to the elliptical structure of the distribution is therefore expected to be unsatisfactory here we propose and analyze a technology that is based on independent component analysis ica we study the proposed icvar methodology in an extensive simulation study and apply it to a high dimensional portfolio situation our analysis yields very accurate vars

nonlinear robin problems with indefinite potential abstract we consider a nonlinear robin problem driven by the p laplacian plus an indefinite potential the conditions on the source term are minimal we prove two multiplicity theorems with sign information for all the solutions in the semilinear case p 2 we show that we can have multiple nodal solutions we apply our results to a special class of logistic equations with equidiffusive reaction

minimax risk over l p balls for l p error consider estimating the mean vector from datan n 2 i withl q norm loss q 1 when is known to lie in ann dimensionall p ball p 0 for largen the ratio of minimaxlinear risk to minimax risk can bearbitrarily large ifp q obvious exceptions aside the limiting ratio equals 1 only ifp q 2 our arguments are mostly indirect involving a reduction to a univariate bayes minimax problem whenp q simple non linear co ordinatewise threshold rules are asymptotically minimax at small signal to noise ratios and within a bounded factor of asymptotic minimaxity in general we also give asymptotic evaluations of the minimax linear risk our results are basic to a theory of estimation in besov spaces using wavelet bases to appear elsewhere

nontrivial solutions of semilinear equations at resonance in this paper we prove new existence results concerning nontrivial solutions to semilinear elliptic problem at resonance the methods used here are based on combining the minimax methods and the morse theory

nontrivial critical points for asymptotically quadratic function abstract in this paper we study the existence of nontrivial critical points as applications we study the existence of nontrivial solutions of the dirichelet problem for nonlinear elliptic pde and periodic boundary value problem for nonlinear ope

on resonant neumann problems we consider semilinear neumann problems at resonance and prove existence and multiplicity theorems the existence theorems allow resonance with respect to any eigenvalue of the negative neumann laplacian the multiplicity theorems concern problems resonant at 0 the principal eigenvalue or at the first nonzero eigenvalue our approach uses tools from critical point theory and from morse theory

non uniformly asymptotically linear p laplacian problems abstract we relax conditions on uniform convergences of f x s s p 2 s which are essential in many results on multiplicity for asymptotically linear p laplacian problems

a saddle point approximation for certain multivariate tail probabilities we consider the problem of computing the probability of a union of a large number of tail events this multivariate probability is intractable so we propose a bound based on univariate and bivariate marginal probabilities which in turn we approximate by the saddle point method and a correction for skewness we use this approach to approximate a reliability index for power systems our numerical results indicate that this deterministic bound is also an accurate estimate and that it is more efficient than direct monte carlo

multiplicity results and qualitative properties for neumann semilinear elliptic problems abstract in this paper we establish the existence of multiple ordered classical solutions for neumann semilinear elliptic problems and provide qualitative information about them no symmetry assumptions are required neither on the nonlinearity nor on the domain for some results the growth of the nonlinearity at infinity is arbitrary we also consider the cases when the nonlinearity is either superlinear or asymptotically linear with strong resonances extensive use is made of the mountain pass theorem and leray schauder topological degree

nontrivial solutions for discrete boundary value problems with multiple resonance via computations of the critical groups abstract in this paper we study the existence of nontrivial solutions for a class of second order difference equations with multiple resonance at both infinity and the origin by applying the critical point theory and morse theory

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

multiple solutions with precise sign for nonlinear parametric robin problems we consider a parametric nonlinear robin problem driven by the p laplacian we show that if the parameter 2 the second eigenvalue of the robin p laplacian then the problem has at least three nontrivial solutions two of constant sign and the third nodal in the semilinear case p 2 we show that we can generate a second nodal solution our approach uses variational methods truncation and perturbation techniques and morse theory in the process we produce two useful remarks about the first two eigenvalues of the robin p laplacian

multiple solutions with sign information for a p 2 equation with combined nonlinearities abstract we consider a parametric nonlinear dirichlet problem driven by the sum of a p laplacian and of a laplacian a p 2 equation and with a reaction which has the competing effects of two distinct nonlinearities a parametric term which is p 1 superlinear convex term and a perturbation which is p 1 sublinear concave term first we show that for all small values of the parameter the problem has at least five nontrivial smooth solutions all with sign information then by strengthening the regularity of the two nonlinearities we produce two more nodal solutions for a total of seven nontrivial smooth solutions all with sign informations our proofs use critical point theory critical groups and flow invariance arguments