Results 1  10
of
127
Adapting to unknown smoothness via wavelet shrinkage
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 1995
"... We attempt to recover a function of unknown smoothness from noisy, sampled data. We introduce a procedure, SureShrink, which suppresses noise by thresholding the empirical wavelet coefficients. The thresholding is adaptive: a threshold level is assigned to each dyadic resolution level by the princip ..."
Abstract

Cited by 679 (19 self)
 Add to MetaCart
We attempt to recover a function of unknown smoothness from noisy, sampled data. We introduce a procedure, SureShrink, which 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 smoothnessadaptive: if the unknown function contains jumps, the reconstruction (essentially) does also; if the unknown function has a smooth piece, the reconstruction is (essentially) as smooth as the mother wavelet will allow. The procedure is in a sense optimally smoothnessadaptive: it is nearminimax 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 smoothing methods  kernels, splines, and orthogonal series estimates  even with optimal choices of the smoothing parameter, would be unable to perform
Minimax Estimation via Wavelet Shrinkage
, 1992
"... 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 coe cients. The shrinkage can be tuned to be nearly minim ..."
Abstract

Cited by 246 (32 self)
 Add to MetaCart
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 coe cients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel and Besovtype 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 our method can signi cantly outperform every linear method (kernel, smoothing spline, sieve,:::) in a minimax sense. Variants of our method based on simple threshold nonlinearities are nearly minimax. Our method possesses the interpretation of spatial adaptivity: it reconstructs using a kernel which mayvary 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 discuss practical implementation, spatial adaptation properties and applications to inverse problems.
Wavelet shrinkage: asymptopia
 Journal of the Royal Statistical Society, Ser. B
, 1995
"... Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators bein ..."
Abstract

Cited by 239 (35 self)
 Add to MetaCart
Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators being obtained for a variety of interesting problems. Unfortunately, the results have often not been translated into practice, for a variety of reasons { sometimes, similarity to known methods, sometimes, computational intractability, and sometimes, lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coe cients towards the origin by an amount p p 2 log(n) = n. The method is di erent from methods in common use today, is computationally practical, and is spatially adaptive; thus it avoids a number of previous objections to minimax estimators. At the same time, the method is nearly minimax for a wide variety of loss functions { e.g. 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 classes, Sobolev classes, and Bounded Variation. This is amuch broader nearoptimality than anything previously proposed in the minimax literature. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and informationbased complexity.
Nonlinear solution of linear inverse problems by waveletvaguelette decomposition
, 1992
"... We describe the WaveletVaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such asnumerical di erentiation, inversion of Abeltype ..."
Abstract

Cited by 183 (12 self)
 Add to MetaCart
We describe the WaveletVaguelette Decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type { such asnumerical di erentiation, inversion of Abeltype transforms, certain convolution transforms, and the Radon Transform. We propose to solve illposed linear inverse problems by nonlinearly \shrinking" the WVD coe cients of the noisy, indirect data. Our approach o ers signi cant advantages over traditional SVD inversion in the case of recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L 2 loss, over the entire Besov scale. The important case of Besov spaces Bp;q, p <2, which model spatial inhomogeneity, is included. In comparison, linear procedures { SVD included { cannot attain optimal rates of convergence over such classes in the case p<2. For example, our methods achieve faster rates of convergence, for objects known to lie in the Bump Algebra or in Bounded Variation, than any linear procedure.
Adaptive wavelet estimation: A block thresholding and oracle inequality approach
 Ann. Statist
, 1999
"... We study wavelet function estimation via the approach of block thresholding 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 ..."
Abstract

Cited by 98 (14 self)
 Add to MetaCart
We study wavelet function estimation via the approach of block thresholding 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 regression is proposed and the optimality of the procedure is investigated. We show that the estimator achieves simultaneously three objectives: adaptivity, 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 implement, at the computational cost of O�n�. Simulation shows that the estimator has excellent numerical performance relative to more traditional wavelet estimators. 1. Introduction. Wavelet
Asymptotic Equivalence of Density Estimation and Gaussian White Noise
 Ann. Statist
, 1996
"... Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam' ..."
Abstract

Cited by 81 (3 self)
 Add to MetaCart
Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance \Delta would make it precise. The models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression (Brown and Low, 1993). We consider the analogous problem for the experiment given by n i. i. d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Holder ball with exponent ff ? 1 2 and and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally asymptotically equivalent to a white noise experiment with dri...
Generalized Likelihood Ratio Statistics And Wilks Phenomenon
, 2000
"... this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are ..."
Abstract

Cited by 77 (20 self)
 Add to MetaCart
this paper. We introduce the generalized likelihood statistics to overcome the drawbacks of nonparametric maximum likelihood ratio statistics. New Wilks phenomenon is unveiled. We demonstrate that a class of the generalized likelihood statistics based on some appropriate nonparametric estimators are asymptotically distribution free and follow
Incorporating Information on Neighboring Coefficients into Wavelet Estimation
, 1999
"... In standard wavelet methods, the empirical wavelet coefficients are thresholded term by term, on the basis of their individual magnitudes. Information on other coefficients has no influence on the treatment of particular coefficients. We propose a wavelet shrinkage method that incorporates informati ..."
Abstract

Cited by 53 (9 self)
 Add to MetaCart
In standard wavelet methods, the empirical wavelet coefficients are thresholded term by term, on the basis of their individual magnitudes. Information on other coefficients has no influence on the treatment of particular coefficients. We propose a wavelet shrinkage method that incorporates information on neighboring coefficients into the decision making. The coefficients are considered in overlapping blocks; the treatment of coefficients in the middle of each block depends on the data in the whole block. The asymptotic and numerical performances of two particular versions of the estimator are investigated. We show that, asymptotically, one version of the estimator achieves the exact optimal rates of convergence over a range of Besov classes for global estimation, and attains adaptive minimax rate for estimating functions at a point. In numerical comparisons with various methods, both versions of the estimator perform excellently.
Recovering Edges in IllPosed Inverse Problems: Optimality of Curvelet Frames
, 2000
"... We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. Traditional linear methods for solving such in ..."
Abstract

Cited by 49 (14 self)
 Add to MetaCart
We consider a model problem of recovering a function f(x1,x2) from noisy Radon data. The function f to be recovered is assumed smooth apart from a discontinuity along a C2 curve – i.e. an edge. We use the continuum white noise model, with noise level ɛ. 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 ɛ only as O(ɛ1/2)asɛ → 0. A recent innovation – nonlinear shrinkage in the wavelet domain – visually improves edge sharpness and improves MSE convergence to O(ɛ2/3). However, as we show here, this rate is not optimal. In fact, essentially optimal performance is obtained by deploying the recentlyintroduced 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 curveletbased biorthogonal decomposition
Rates of Convergence of Posterior Distributions
, 1998
"... We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are driven by two quantities: the size of the space, as measure by metric entropy or bracketing entropy, and ..."
Abstract

Cited by 48 (0 self)
 Add to MetaCart
We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are driven by two quantities: the size of the space, as measure by metric entropy or bracketing entropy, and the degree to which the prior concentrates in a small ball around the true parameter. We apply the results to several examples. In some cases, natural priors give suboptimal rates of convergence and better rates can be obtained by using sievebased priors such as those introduced by Zhao (1993, 1998). AMS 1990 classification: Primary, 62A15, Secondary: 62E20, 62G15. KEYWORDS: Bayesian inference, asymptotic inference, nonparametric models, sieves. 1 Introduction. Nonparametric Bayesian methods have become quite popular lately, largely because of advances in computing; see Dey, Mueller and Sinha (1998) for a recent account. Because of their growing popularity, it is important to understand ...