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

a wavelet shrinkage approach to tomographic image reconstruction abstract a method is proposed for reconstructing images from tomographic data with respect to a two dimensional wavelet basis the wavelet vaguelette decomposition wvd is used as a framework within which expressions for the necessary wavelet coefficients may be derived these coefficients are calculated using a version of the filtered back projection algorithm as a computational tool in a multiresolution fashion the necessary filters are defined in terms of the underlying wavelets denoising is accomplished through an adaptation of the wavelet shrinkage ws approach of donoho et al and amounts to a form of regularization combining these two steps yields the proposed wvd ws reconstruction algorithm which is compared to the traditional filtered backprojection method in a small study

on local smoothing of nonparametric curve estimators abstract we develop new local versions of familiar smoothing methods such as cross validation and smoothed cross validation in the contexts of density estimation and regression these new methods are locally adaptive in the sense that they capture smooth local fluctuations in the curve by using smoothly varying bandwidths that change as the character of the curve changes moreover the new methods are accurate easy to apply and computationally expedient

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

jump and sharp cusp detection by wavelets summary a method is proposed to detect jumps and sharp cusps in a function which is observed with noise by checking if the wavelet transformation of the data has significantly large absolute values across fine scale levels asymptotic theory is established and practical implementation is discussed the method is tested on simulated examples and applied to stock market return data the analysis of change points which describe sudden localised changes has recently found increasing interest change points can be used to model practical problems arising in fields such as quality control economics medicine signal and image processing and physical sciences for example in electroencephalogram signals sharp cusps exhibit the accelerations and decelerations in the beating of the hearts many practical problems like this involve functions which have jumps and sharp cusps the recently developed theory of wavelets has drawn much attention from both math ematicians statisticians and engineers in the seminal work of donoho 1993 donoho johnstone 1994 1995a b and donoho johnstone et al 1995 orthonormal bases of compactly supported wavelets have been used to estimate functions the theory of wavelets permits decomposition of functions into localised oscillating components this is an ideal tool to study localised changes such as jumps and sharp cusps in one dimension as well as several dimensions unlike traditional smoothing methods based on a fixed spatial scale the wavelet method is a multiresolution approach and has local adaptivity in this

vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices this work provides a general formulation to solve vibration problems for continuous systems with damping effects including modal transient harmonic and spectrum response analyses in modal analysis the system eigenvalues and corresponding eigenfunctions can be determined the orthogonal relations of eigenfunctions are shown for transient harmonic and spectrum analyses the generic force actuator functions and response sensing operators are introduced respectively and used to derive the system response the time domain response is obtained for transient analysis the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis the solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force actuator and response sensor mode shape functions in particular one dimensional beam and two dimensional plate vibration analyses are illustrated by following the developed generic formulation this work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well

frequentist optimality of bayesian wavelet shrinkage rules for gaussian and non gaussian noise the present paper investigates theoretical performance of various bayesian wavelet shrinkage rules in a nonparametric regression model with i i d errors which are not necessarily normally distributed the main purpose is comparison of various bayesian models in terms of their frequentist asymptotic optimality in sobolev and besov spaces we establish a relationship between hyperparameters verify that the majority of bayesian models studied so far achieve theoretical optimality state which bayesian models cannot achieve optimal convergence rate and explain why it happens

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

an essay on statistical decision theory 2000 an essay on statistical decision theory journal of the american statistical association vol 95 no 452 pp 1277 1281

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

capacity and power allocation for fading mimo channels with channel estimation error in this correspondence we investigate the effect of channel estimation error on the capacity of multiple input multiple output mimo fading channels we study lower and upper bounds of mutual information under channel estimation error and show that the two bounds are tight for gaussian inputs assuming gaussian inputs we also derive tight lower bounds of ergodic and outage capacities and optimal transmitter power allocation strategies that achieve the bounds under perfect feedback for the ergodic capacity the optimal strategy is a modified waterfilling over the spatial antenna and temporal fading domains this strategy is close to optimum under small feedback delays but when the delay is large equal powers should be allocated across spatial dimensions for the outage capacity the optimal scheme is a spatial waterfilling and temporal truncated channel inversion numerical results show that some capacity gain is obtained by spatial power allocation temporal power adaptation on the other hand gives negligible gain in terms of ergodic capacity but greatly enhances outage performance

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

empirical bayes approach to improve wavelet thresholding for image noise reduction wavelet threshold algorithms replace small magnitude wavelet coefficients with zero and keep or shrink the other coefficients this is basically a local procedure because wavelet coefficients characterize the local regularity of a function although a wavelet transform has decorrelating properties structures in images like edges are never decorrelated completely and these structures appear in the wavelet coefficients a classification based on a local criterion like coefficient magnitude is not the perfect method to distinguish important uncorrupted coefficients from coefficients dominated by noise we therefore introduce a geometrical prior model for configurations of important wavelet coefficients and combine this with local characterization of a classical threshold procedure into a bayesian framework the local characterization is incorporated into the conditional model whereas the prior model describes only configurations not coefficient values more precisely local characterization favors co

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

general empirical bayes wavelet methods and exactly adaptive minimax estimation in many statistical problems stochastic signals can be represented as a sequence of noisy wavelet coefficients in this paper we develop general empirical bayes methods for the estimation of true signal our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals in particular our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in besov balls of all smoothness and shape indices and they are uniformly superefficient in convergence rates in all compact sets in besov spaces with a finite secondary shape parameter furthermore in classes nested between besov balls of the same smoothness index our estimators dominate threshold and james stein estimators within an infinitesimal fraction of the minimax risks more general block empirical bayes estimators are developed both white noise with drift and nonparametric regression are considered

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

stochastic expansions in an overcomplete wavelet dictionary we consider random functions defined in terms of members of an overcomplete wavelet dictionary the function is modelled as a sum of wavelet components at arbitrary positions and scales where the locations of the wavelet components and the magnitudes of their coefficients are chosen with respect to a marked poisson process model the relationships between the parameters of the model and the parameters of those besov spaces within which realizations will fall are investigated the models allow functions with specified regularity properties to be generated they can potentially be used as priors in a bayesian approach to curve estimation extending current standard wavelet methods to be free from the dyadic positions and scales of the basis functions

wedgelets nearly minimax estimation of edges we study a simple horizon model for the problem of recovering an image from noisy data in this model the image has an edge with fi holder regularity adopting the viewpoint of computational harmonic analysis we develop an overcomplete collection of atoms called wedgelets dyadically organized indicator functions with a variety of locations scales and orientations the wedgelet representation provides nearly optimal representations of objects in the horizon model as measured by minimax description length we show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet decorated recursive partition which minimizes a complexity penalized sum of squares this estimate using su cient sub pixel resolution achieves nearly the minimax mean squared error in the horizon model in fact the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon 1 fi 2 wedgelet analysis and de noising may be used successfully outside the horizon model we study images modelled as indicators of star shaped sets with smooth bound aries and show that complexity penalized wedgelet partitioning achieves nearly the minimax risk in that setting also

on feasibility of interference alignment in mimo interference networks we explore the feasibility of interference alignment in signal vector space based only on beamforming for k user mimo interference channels our main contribution is to relate the feasibility issue to the problem of determining the solvability of a multivariate polynomial system which is considered extensively in algebraic geometry it is well known e g from bezout s theorem that generic polynomial systems are solvable if and only if the number of equations does not exceed the number of variables following this intuition we classify signal space interference alignment problems as either proper or improper based on the number of equations and variables rigorous connections between feasible and proper systems are made through bernshtein s theorem for the case where each transmitter uses only one beamforming vector the multibeam case introduces dependencies among the coefficients of a polynomial system so that the system is no longer generic in the sense required by both theorems in this case we show that the connection between feasible and proper systems can be further strengthened since the equivalency between feasible and proper systems does not always hold by including standard information theoretic outer bounds in the feasibility analysis

adaptive estimation of the intensity of inhomogeneous poisson processes via concentration inequalities in this paper we establish oracle inequalities for penalized projection estimators of the intensity of an inhomogeneous poisson process we study consequently the adaptive properties of penalized projection estimators at first we provide lower bounds for the minimax risk over various sets of smoothness for the intensity and then we prove that our estimators achieve these lower bounds up to some constants the crucial tools to obtain the oracle inequalities are new concentration inequalities for suprema of integral functionals of poisson processes which are analogous to talagrand s inequalities for empirical processes

does median filtering truly preserve edges better than linear filtering image processing researchers commonly assert that median filtering is better than linear filtering for removing noise in the presence of edges using a straightforward large n decision theory framework this folk theorem is seen to be false in general we show that median filtering and linear filtering have similar asymptotic worst case mean squared error mse when the signal to noise ratio snr is of order 1 which corresponds to the case of constant per pixel noise level in a digital signal to see dramatic benefits of median smoothing in an asymptotic setting the per pixel noise level should tend to zero i e snr should grow very large we show that a two stage median filtering using two very different window widths can dramatically outperform traditional linear and median filtering in settings where the underlying object has edges in this two stage procedure the first pass at a fine scale aims at increasing the snr the second pass at a coarser scale correctly exploits the nonlinearity of the median image processing methods based on nonlinear partial differential equations pdes are often said to improve on linear filtering in the presence of edges such methods seem difficult to analyze rigorously in a decision theoretic framework a popular example is mean curvature motion mcm which is formally a kind of iterated median filtering our results on iterated median filtering suggest that some pde based methods are candidates to rigorously outperform linear filtering in an asymptotic framework

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

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

computational and statistical boundaries for submatrix localization in a large noisy matrix we study in this paper computational and statistical boundaries for submatrix localization given one observation of one or multiple nonoverlapping signal submatrix of magnitude lambda and size k m times k n embedded in a large noise matrix of size m times n the goal is to optimal identify the support of the signal submatrix computationally and statistically two transition thresholds for the signal to noise ratio lambda sigma are established in terms of m n k m and k n the first threshold sf snr c corresponds to the computational boundary we introduce a new linear time spectral algorithm that identifies the submatrix with high probability when the signal strength is above the threshold sf snr c below this threshold it is shown that no polynomial time algorithm can succeed in identifying the submatrix under the hidden clique hypothesis the second threshold sf snr s captures the statistical boundary below which no method can succeed in localization with probability going to one in the minimax sense the exhaustive search method successfully finds the submatrix above this threshold in marked contrast to submatrix detection and sparse pca the results show an interesting phenomenon that sf snr c is always significantly larger than sf snr s under the sub gaussian error model which implies an essential gap between statistical optimality and computational efficiency for submatrix localization

statistical inverse estimation in hilbert scales the recovery of signals from indirect measurements blurred by random noise is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable for greater flexibility the general problem is embedded in an abstract hilbert scale in the applications sobolev scales are used for the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product where the operator is bounded but not necessarily compact a lower bound to certain minimax rates is included and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate examples include errors in variables deconvolution and indirect nonparametric regression special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation

universal algorithms for learning theory part i piecewise constant functions this paper is concerned with the construction and analysis of a universal estimator for the regression problem in supervised learning universal means that the estimator does not depend on any a priori assumptions about the regression function to be estimated the universal estimator studied in this paper consists of a least square fitting procedure using piecewise constant functions on a partition which depends adaptively on the data the partition is generated by a splitting procedure which differs from those used in cart algorithms it is proven that this estimator performs at the optimal convergence rate for a wide class of priors on the regression function namely as will be made precise in the text if the regression function is in any one of a certain class of approximation spaces or smoothness spaces of order not exceeding one a limitation resulting because the estimator uses piecewise constants measured relative to the marginal measure then the estimator converges to the regression function in the least squares sense with an optimal rate of convergence in terms of the number of samples the estimator is also numerically feasible and can be implemented on line

covariance structure of wavelet coefficients theory and models in a bayesian perspective we present theoretical results on the random wavelet coefficients covariance structure we use simple properties of the coefficients to derive a recursive way to compute the within and across scale covariances we point out a usetul link between the algorithm proposed and the two dimensional discrete wavelel transform we then focus on bayesian wavelet shrinkage for estimating a function from noisy data a prior distribution is imposed on the coefficients of the unknown function we show how our findings on the covariance structure make it possible to specify priors that take into account the full correlation between coefficients through a parsimonious number of hyperparameters we use markov chain monte carlo rnethods to estirnate the parameters and illustrate our method on bench mark simulated signals

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

linear interference alignment in full duplex mimo networks with imperfect csi in this paper we consider a system where full duplex fd base stations bss communicate with half duplex hd downlink dl and uplink ul users in a multi user multi cell network where all nodes are equipped with multiple antennas the introduction of fd bss offers potential to increase spectral efficiency however it also causes a surge in the number of interference links compared with the hd network counterpart here we apply linear interference alignment ia to manage interference in this network under imperfect channel state information csi first we characterize the performance losses incurred with respect to the achievable sum rate and degrees of freedom dofs results show that the general trend in performance loss is mainly determined by how the error scales with the signal to noise ratio snr in particular full ul and dl dof can be achieved even under imperfect csi when the channel error is at least inversely proportional to snr moreover in such cases the sum rate loss is always finite and either goes to zero or is upper bounded by a derived value second we design two linear ia algorithms applicable to the system under consideration these are based on minimizing the mean square error mmse and maximizing the signal to interference plus noise ratio and consider statistical knowledge of the csi error for added robustness the proposed algorithms follow specific design principles that distribute the different interference components amongst the various beamformers and result in unitary receivers and precoders in addition we show that under certain conditions both designs result in identical beamforming solutions even though the mmse algorithm has lower computational complexity finally we also derive the proper condition for ia feasibility in the multi cell system under consideration

dimension reduction for hypothesis testing in worst case scenarios this paper considers a one among many detection problem where one has to discriminate between pure noise and one among l alternatives that are known up to an amplitude factor two issues linked to high dimensionality arise in this framework first the computational complexity associated to the generalized likelihood ratio glr with the constraint of sparsity one inflates linearly with l which can be an obstacle when multiple data sets have to be tested second standard procedures based on dictionary learning aimed at reducing the dimensionality may suffer from severe power losses for some alternatives thus suggesting a worst case scenario strategy in the case where the learned dictionary has k 1 column we show that the exact solution of the resulting detection problem which can be formulated as a minimax problem can be obtained by quadratic programming because it allows a better sampling of the diversity of the alternatives the case k 1 is expected to improve the detection performances over the case k 1 the worst case analysis of this case which is more involved leads us to propose two minimax learning algorithms numerical results show that these algorithms indeed allow to increase performances over the k 1 case and are in fact comparable to the glr using the full set of alternatives while being computationally simpler

on deriving the inverse of a sum of matrices available expressions are reviewed and new ones derived for the inverse of the sum of two matrices one of them being nonsingular particular attention is given to text bf a text bf ubv 1 where textbf a is nonsingular and textbf u b and textbf v may be rectangular generalized inverses of textbf a textbf ubv are also considered several statistical applications are discussed

why simple shrinkage is still relevant for redundant representations shrinkage is a well known and appealing denoising technique introduced originally by donoho and johnstone in 1994 the use of shrinkage for denoising is known to be optimal for gaussian white noise provided that the sparsity on the signal s representation is enforced using a unitary transform still shrinkage is also practiced with nonunitary and even redundant representations typically leading to very satisfactory results in this correspondence we shed some light on this behavior the main argument in this work is that such simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising bpdn problem while the desired solution of bpdn is hard to obtain in general we develop a simple iterative procedure for the bpdn minimization that amounts to stepwise shrinkage we demonstrate how the simple shrinkage emerges as the first iteration of this novel algorithm furthermore we show how shrinkage can be iterated turning into an effective algorithm that minimizes the bpdn via simple shrinkage steps in order to further strengthen the denoising effect

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

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

a data driven block thresholding approach to wavelet estimation a data driven block thresholding procedure for wavelet regression is proposed and its theoretical and numerical properties are investigated the procedure empirically chooses the block size and threshold level at each resolution level by minimizing stein s unbiased risk estimate the estimator is sharp adaptive over a class of besov bodies and achieves simultaneously within a small constant factor of the minimax risk over a wide collection of besov bodies including both the dense and sparse cases the procedure is easy to implement numerical results show that it has superior finite sample performance in comparison to the other leading wavelet thresholding estimators

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

unbiased estimation of a sparse vector in white gaussian noise the problem studied in this paper is unbiased estimation of a sparse nonrandom vector corrupted by additive white gaussian noise it is shown that while there are infinitely many unbiased estimators for this problem none of them has uniformly minimum variance therefore the focus is placed on locally minimum variance unbiased lmvu estimators simple closed form lower and upper bounds on the variance of lmvu estimators or equivalently on the barankin bound bb are derived these bounds allow an estimation of the threshold region separating the low signal to noise ratio snr and high snr regimes and they indicate the asymptotic behavior of the bb at high snr in addition numerical lower and upper bounds are derived these are tighter than the closed form bounds and thus characterize the bb more accurately numerical studies compare the proposed characterizations of the bb with established biased estimation schemes and demonstrate that while unbiased estimators perform poorly at low snr they may perform better than biased estimators at high snr an interesting conclusion of this analysis is that the high snr behavior of the bb depends solely on the value of the smallest nonzero entry of the sparse vector and that this type of dependence is also exhibited by the performance of certain practical estimators

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

forward fourier wavelet regularized deconvolution for ill conditioned systems we propose an efficient hybrid fourier wavelet regularized deconvolution forward algorithm that performs noise regularization via scalar shrinkage in both the fourier and wavelet domains the fourier shrinkage exploits the fourier transform s economical representation of the colored noise inherent in deconvolution whereas the wavelet shrinkage exploits the wavelet domain s economical representation of piecewise smooth signals and images we derive the optimal balance between the amount of fourier and wavelet regularization by optimizing an approximate mean squared error mse metric and find that signals with more economical wavelet representations require less fourier shrinkage forward is applicable to all ill conditioned deconvolution problems unlike the purely wavelet based wavelet vaguelette deconvolution wvd moreover its estimate features minimal ringing unlike the purely fourier based wiener deconvolution even in problems for which the wvd was designed we prove that forward s mse decays with the optimal wvd rate as the number of samples increases further we demonstrate that over a wide range of practical sample lengths forward improves on wvd s performance

free vibrations of a rectangular plate carrying a distributed mass in this paper an analytical method is presented to find the eigenfrequencies of a rectangular plate carrying a uniformly distributed mass using the standard galerkin procedure the equation of motion is reduced to a set of ordinary differential equations from this set the frequency equation is obtained this polynomial equation is solved numerically due to the significance of the fundamental frequency of the system its variation with respect to the non dimensional parameters associated with the location the area density and the distribution area of the mass attached to the plate is investigated furthermore it is shown by a numerical example that the method can be used to study plates with concentrated mass as a special case finally an analysis to obtain the modal surfaces and the related nodal lines is carried out it is demonstrated that the location of the attachment significantly affects the nodal lines and modal interchanges may occur

adaptive thresholding for sparse covariance matrix estimation in this article we consider estimation of sparse covariance matrices and propose a thresholding procedure that is adaptive to the variability of individual entries the estimators are fully data driven and demonstrate excellent performance both theoretically and numerically it is shown that the estimators adaptively achieve the optimal rate of convergence over a large class of sparse covariance matrices under the spectral norm in contrast the commonly used universal thresholding estimators are shown to be suboptimal over the same parameter spaces support recovery is discussed as well the adaptive thresholding estimators are easy to implement the numerical performance of the estimators is studied using both simulated and real data simulation results demonstrate that the adaptive thresholding estimators uniformly outperform the universal thresholding estimators the method is also illustrated in an analysis on a dataset from a small round blue cell tumor microarray experiment a supplement to this ar

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

interference alignment under limited feedback for mimo interference channels this paper analyzes multiple input multiple output interference channels where each receiver knows its channels from all the transmitters and feeds back this information using a limited number of bits to all the other terminals it is shown that as long as the feedback bit rate scales sufficiently fast with the signal to noise ratio the transmitters can use an interference alignment strategy by treating the quantized channel estimates as being perfect to achieve the sum degrees of freedom of the interference channel attainable with perfect and global channel state information a tradeoff between the feedback rate and the achievable degrees of freedom is established by showing that a slower scaling of feedback rate for any one user leads to commensurately fewer degrees of freedom for that user alone it is then shown that under the same fixed transmission strategy but with random quantization the above mentioned sufficient condition on the feedback scaling rate to attain a given sum degrees of freedom up to the maximum attainable is also necessary in this setting

statistical inference for the optimal approximating model in the setting of high dimensional linear models with gaussian noise we investigate the possibility of confidence statements connected to model selection although there exist numerous procedures for adaptive point estimation the construction of adaptive confidence regions is severely limited cf li in ann stat 17 1001 1008 1989 the present paper sheds new light on this gap we develop exact and adaptive confidence regions for the best approximating model in terms of risk one of our constructions is based on a multiscale procedure and a particular coupling argument utilizing exponential inequalities for noncentral 2 distributions we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one

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

recovering edges in ill posed inverse problems optimality of curvelet frames we consider a model problem of recovering a function f x 1 x 2 from noisy radon data the function f to be recovered is assumed smooth apart from a discontinuity along a c 2 curve that is an edge we use the continuum white noise model with noise level varepsilon traditional linear methods for solving such inverse problems behave poorly in the presence of edges qualitatively the reconstructions are blurred near the edges quantitatively they give in our model mean squared errors mses that tend to zero with noise level varepsilon only as o varepsilon 1 2 as varepsilon to 0 a recent innovation nonlinear shrinkage in the wavelet domain visually improves edge sharpness and improves mse convergence to o varepsilon 2 3 however as we show here this rate is not optimal in fact essentially optimal performance is obtained by deploying the recently introduced tight frames of curvelets in this setting curvelets are smooth highly anisotropic elements ideally suited for detecting and synthesizing curved edges to deploy them in the radon setting we construct a curvelet based biorthogonal decomposition of the radon operator and build curvelet shrinkage estimators based on thresholding of the noisy curvelet coefficients in effect the estimator detects edges at certain locations and orientations in the radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain we prove that the curvelet shrinkage can be tuned so that the estimator will attain within logarithmic factors the mse o varepsilon 4 5 as noise level varepsilon to 0 this rate of convergence holds uniformly over a class of functions which are c 2 except for discontinuities along c 2 curves and except for log terms is the minimax rate for that class our approach is an instance of a general strategy which should apply in other inverse problems we sketch a deconvolution example

adaptive estimation in autoregression or mixing regression via model selection we study the problem of estimatingsome unknown regression function in a beta mixing dependent framework to this end we consider some collection of models which are finite dimensional spaces a penalized least squares estimator plse is built on a data driven selected model among this collection we state non asymptotic risk bounds for this plse and give several examples where the procedure can be applied autoregression regression with arithmetically beta mixing design points regression with mixing errors estimation in additive frameworks estimation of the order of the autoregression in addition we show that under a weak moment condition on the errors our estimator is adaptive in the minimax sense simultaneously over some family of besov balls

adaptive lasso for cox s proportional hazards model summary we investigate the variable selection problem for cox s proportional hazards model and propose a unified model selection and estimation procedure with desired theoretical properties and computational convenience the new method is based on a penalized log partial likelihood with the adaptively weighted l1 penalty on regression coefficients providing what we call the adaptive lasso estimator the method incorporates different penalties for different coefficients unimportant variables receive larger penalties than important ones so that important variables tend to be retained in the selection process whereas unimportant variables are more likely to be dropped theoretical properties such as consistency and rate of convergence of the estimator are studied we also show that with proper choice of regularization parameters the proposed estimator has the oracle properties the convex optimization nature of the method leads to an efficient algorithm both simulated and real examples show that the method performs competitively

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

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

on the feasibility of linear interference alignment for mimo interference broadcast channels with constant coefficients in this paper we analyze the feasibility of linear interference alignment ia for multi input multi output mimo interference broadcast channel mimo ibc with constant coefficients we pose and prove the necessary conditions of linear ia feasibility for general mimo ibc except for the proper condition we find another necessary condition to ensure a kind of irreducible interference to be eliminated we then prove the necessary and sufficient conditions for a special class of mimo ibc where the numbers of antennas are divisible by the number of data streams per user since finding an invertible jacobian matrix is crucial for the sufficiency proof we first analyze the impact of sparse structure and repeated structure of the jacobian matrix considering that for the mimo ibc the sub matrices of the jacobian matrix corresponding to the transmit and receive matrices have different repeated structure we find an invertible jacobian matrix by constructing the two sub matrices separately we show that for the mimo ibc where each user has one desired data stream a proper system is feasible for symmetric mimo ibc we provide proper but infeasible region of antenna configurations by analyzing the difference between the necessary conditions and the sufficient conditions of linear ia feasibility

the dfs fused lasso linear time denoising over general graphs the fused lasso also known as anisotropic total variation denoising is widely used for piecewise constant signal estimation with respect to a given undirected graph the fused lasso estimate is highly nontrivial to compute when the underlying graph is large and has an arbitrary structure but for a special graph structure namely the chain graph the fused lasso or simply 1d fused lasso can be computed in linear time in this paper we revisit a result recently established in the online classification literature herbster et al 2009 cesa bianchi et al 2013 and show that it has important implications for signal denoising on graphs the result can be translated to our setting as follows given a general graph if we run the standard depth first search dfs traversal algorithm then the total variation of any signal over the chain graph induced by dfs is no more than twice its total variation over the original graph this result leads to several interesting theoretical and computational conclusions letting m and n denote the number of edges and nodes respectively of the graph in consideration it implies that for an underlying signal with total variation t over the graph the fused lasso properly tuned achieves a mean squared error rate of t2 3n 2 3 moreover precisely the same mean squared error rate is achieved by running the 1d fused lasso on the dfs induced chain graph importantly the latter estimator is simple and computationally cheap requiring o m operations to construct the dfs induced chain and o n operations to compute the 1d fused lasso solution over this chain further for trees that have bounded maximum degree the error rate of t2 3n 2 3 cannot be improved in the sense that it is the minimax rate for signals that have total variation t over the tree finally several related results also hold for example the analogous result holds for a roughness measure defined by the l0 norm of differences across edges in place of the total variation metric

on vector and matrix median computation the aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them we prove relations between vector and matrix means and medians particularly regarding the classical structure tensor moreover we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2i 2 matrices which frequently arise in image processing our findings are explained and illustrated by numerical examples to solve the corresponding minimization problems we propose several algorithms existing approaches include weiszfeld s algorithm for the computation of 2 vector medians and semi definite programming in particular second order cone programming which has been used for matrix median computation in this paper we adapt weiszfeld s algorithm for our setting and show that also two splitting methods namely the alternating direction method of multipliers and the parallel proximal algorithm can be applied for generalized vector and matrix median computations besides we compare the performance of these algorithms numerically and apply them within local median filters

on the degrees of freedom achievable through interference alignment in a mimo interference channel consider a k user flat fading mimo interference channel where the kth transmitter or receiver is equipped with mk respectively nk antennas if an exponential in k number of generic channel extensions are used either across time or frequency cadambe and jafar 1 showed that the total achievable degrees of freedom dof can be maximized via interference alignment resulting in a total dof that grows linearly with a even if mk and nk are bounded in this work we consider the case where no channel extension is allowed and establish a general condition that must be satisfied by any degrees of freedom tuple d1 d2 dk achievable through linear interference alignment for a symmetric system with mk m nk n dk d for all k this condition implies that the total achievable dof cannot grow linearly with k and is in fact no more than k m n k 1 we also show that this bound is tight when the number of antennas at each transceiver is divisible by d the number of data streams per user

signal denoising and ultrasonic flaw detection via overcomplete and sparse representations sparse signal representations from overcomplete dictionaries are the most recent technique in the signal processing community applications of this technique extend into many fields in this paper this technique is utilized to cope with ultrasonic flaw detection and noise suppression problem in particular a noisy ultrasonic signal is decomposed into sparse representations using a sparse bayesian learning algorithm and an overcomplete dictionary customized from a gabor dictionary by incorporating some a priori information of the transducer used nonlinear postprocessing including thresholding and pruning is then applied to the decomposed coefficients to reduce the noise contribution and extract the flaw information because of the high compact essence of sparse representations flaw echoes are packed into a few significant coefficients and noise energy is likely scattered all over the dictionary atoms generating insignificant coefficients this property greatly increases the efficiency of the pruning and thresholding operations and is extremely useful for detecting flaw echoes embedded in background noise the performance of the proposed approach is verified experimentally and compared with the wavelet transform signal processor experimental results to detect ultrasonic flaw echoes contaminated by white gaussian additive noise or correlated noise are presented in the paper

performance analysis of ia techniques in the mimo ibc with imperfect csi in this work we consider the multiple input multiple output mimo interference broadcast channel ibc and analyse the performance of interference alignment ia under imperfect channel state information csi where the variance of the csi error depends on the signal to noise ratio snr first we derive an upper bound on asymptotic mean loss in sum rate compared to the perfect csi case and then we quantify the achievable degrees of freedom dof with imperfect csi both sum rate loss and achievable dof are found to be dependent on the number of cells in the system and the amount of users per cell in addition to the csi error parameters themselves results show that when error variance is inversely proportional to snr full dof are achievable and the asymptotic sum rate loss is bounded by a derived value additionally if the csi imperfection does not disappear for asymptotically high snr then the full dof gain promised by ia cannot be achieved we quantify this loss in relation to the csi mismatch itself the analytically derived bounds are validated via system simulation with the cellular counterparts of the maximum signal to interference plus noise ratio max sinr and the minimum weighted leakage interference min wli algorithms being the ia techniques of choice secondly inspired by the csi mismatch model used to derive the bounds we present a novel max sinr algorithm with stochastic csi error knowledge max sinr scek for the mimo ibc simulations show that the proposed algorithm improves performance over the standard one under imperfect csi conditions without any additional computational costs

formulae for mean integrated squared error of nonlinear wavelet based density estimators we provide an asymptotic formula for the mean integrated squared error mise of nonlinear wavelet based density estimators we show that unlike the analogous situation for kernel density estimators this mise formula is relatively unaffected by assumptions of continuity in particular it is available for densities which are smooth in only a piecewise sense another difference is that in the wavelet case the classical mise formula is valid only for sufficiently small values of the bandwidth for larger bandwidths mise assumes a very different form and hardly varies at all with changing bandwidth this remarkable property guarantees a high level of robustness against oversmoothing not encountered in the context of kernel methods we also use the mise formula to describe an asymptotically optimal empirical bandwidth selection rule

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

the root unroot algorithm for density estimation as implemented via wavelet block thresholding we propose and implement a density estimation procedure which begins by turning density estimation into a nonparametric regression problem this regression problem is created by binning the original observations into many small size bins and by then applying a suitable form of root transformation to the binned data counts in principle many common nonparametric regression estimators could then be applied to the transformed data we propose use of a wavelet block thresholding estimator in this paper finally the estimated regression function is un rooted by squaring and normalizing the density estimation procedure achieves simultaneously three objectives computational efficiency adaptivity and spatial adaptivity a numerical example and a practical data example are discussed to illustrate and explain the use of this procedure theoretically it is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the besov classes the estimator also automatically adapts to the local smoothness of the underlying function and attains the local adaptive minimax rate for estimating functions at a point there are three key steps in the technical argument poissonization quantile coupling and oracle risk bound for block thresholding in the non gaussian setting some of the technical results may be of independent interest

empirical volatility analysis feature detection and signal extraction with function dictionaries abstract we aim to investigate the potential usefulness of wavelets for representing and decomposing financial volatility processes our strategy relies on the empirical analysis of high frequency intradaily stock index returns by using adaptive signal processing techniques which exploit the approximation and computational power of wavelet transforms we first deal with data pre processing and pre smoothing before addressing the statistical model building stage we thus introduce a flexible parametric model that yields an effective empirical volatility analysis tool capable of handling and detecting latent periodicities and consequently delivering more accurate signal estimates we extract the structure of volatility through the information content of projected signals obtained by representing and approximating the observed returns with special function dictionaries that may significantly contribute to reduce the risk that standard volatility models might fail to achieve meaningful statistical inference

use of the analytical and numerical combined method in the free vibration analysis of a rectangular plate with any number of point masses and translational springs abstract by means of the analytical and numerical combined method ancm the natural frequencies and the corresponding mode shapes of a uniform rectangular flat plate carrying any number of point masses and translational springs are determined the boundary supported conditions of the plate and the magnitudes and locations of the concentrated elements are arbitrary first the closed form solution of the natural frequencies and the normal mode shapes of the unconstrained plate without any concentrated elements attached are obtained second based on the closed form solution and using the mode superposition technique the eigenvalue equation of the constrained plate with any number of concentrated elements attached is derived finally the eigenvalue equation is solved numerically to give the desired natural frequencies and mode shapes of the constrained plate under the condition that the accuracies of the sought natural frequencies are approximately the same the order of the eigenvalue equation derived from the ancm is much lower than that derived from the traditional finite element method fem thus the cpu time required by the ancm is much less than that required by the fem furthermore the ancm is also superior to the pure analytical closed form solution method since the former ancm is available for free vibration analysis of a uniform rectangular flat plate carrying any number of concentrated elements but the latter is usually practical only for single concentrated elements

minimax adaptive spectral estimation from an ensemble of signals we develop a statistical method for estimating the spectrum from a data set that consists of several signals all of which are realizations of a common random process we first find estimates of the common spectrum using each signal then we construct m partial aggregates each partial aggregate is a linear combination of m 1 of the spectral estimates the weights are obtained from the data via a least squares criterion the final spectral estimate is the average of these m partial aggregates we show that our final estimator is minimax rate adaptive if at least two of the estimators per signal attain the optimal rate n 2alpha 2alpha 1 for spectra belonging to a generalized lipschitz ball with smoothness index alpha our simulation study strongly suggests that our procedure works well in practice and in a large variety of situations is preferable to the simple averaging of the m spectral estimates

adaptive de noising of signals satisfying differential inequalities the paper is devoted to spatial adaptive estimation of signals satisfying linear differential inequalities with an unknown differential operator of a given order the classes of signals under consideration cover a wide variety of classes common to nonparametric regression in particular they contain the signals whose parameters of smoothness are not uniformly bounded even locally we develop an estimator which is optimal in order over a wide range of the classes and discrete global accuracy measures

use of moving distributed mass element for the dynamic analysis of a flat plate undergoing a moving distributed load this paper presents the theory regarding a moving distributed mass element so that the dynamic responses of a rectangular plate subjected to a moving distributed mass with the effects of inertia force coriolis force and centrifugal force considered can be easily determined in which the property matrices of the moving distributed mass element are derived by means of the principle of superposition and the definition of shape functions and the overall property matrices of the entire vibrating system are determined from the combination of the last element property matrices of the moving distributed mass element and those of the plate itself since the property matrices of the moving distributed mass element have something to do with the instantaneous position and the distribution of the moving mass they are time dependent and so are the overall property matrices of the entire vibrating system based on the last concept the equations of motion for a rectangular plate subjected to a moving distributed mass are established and solved to yield the dynamic responses of the entire vibrating system numerical results reveal that the factors such as the contact area between the moving distributed mass and the plate moving load speed inertia force coriolis force and centrifugal force affect the vibration characteristics of the plate to some degree copyright 2006 john wiley sons ltd

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