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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 ..."
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Cited by 990 (20 self)
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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 ..."
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Cited by 322 (32 self)
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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 ..."
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Cited by 297 (36 self)
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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.
Density estimation by wavelet thresholding
 Ann. Statist
, 1996
"... Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coe cients. Minimax rates of convergence are studied over a large range of Besov function classes Bs;p;q and for a rang ..."
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Cited by 208 (8 self)
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Density estimation is a commonly used test case for nonparametric estimation methods. We explore the asymptotic properties of estimators based on thresholding of empirical wavelet coe cients. Minimax rates of convergence are studied over a large range of Besov function classes Bs;p;q and for a range of global L 0 p error measures, 1 p 0 < 1. A single wavelet threshold estimator is asymptotically minimax within logarithmic terms simultaneously over a range of spaces and error measures. In particular, when p 0> p, some form of nonlinearity is essential, since the minimax linear estimators are suboptimal by polynomial powers of n. A second approach, using an approximation of a Gaussian white noise model in a Mallows metric, is used to attain exactly optimal rates of convergence for quadratic error (p 0 = 2).
Unconditional bases are optimal bases for data compression and for statistical estimation
 Applied and Computational Harmonic Analysis
, 1993
"... An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and ..."
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Cited by 172 (21 self)
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An orthogonal basis of L 2 which is also an unconditional basis of a functional space F is a kind of optimal basis for compressing, estimating, and recovering functions in F. Simple thresholding operations, applied in the unconditional basis, work essentially better for compressing, estimating, and recovering than they do in any other orthogonal basis. In fact, simple thresholding in an unconditional basis works essentially better for recovery and estimation than other methods, period. (Performance is measured in an asymptotic minimax sense.) As an application, we formalize and prove Mallat's Heuristic, which says that wavelet bases are optimal for representing functions containing singularities, when there may be an arbitrary number of singularities, arbitrarily distributed.
Adaptation To High Spatial Inhomogeneity Using Wavelet Methods
, 1999
"... Many of the signals to which wavelet methods are applied, including those encountered in simulation experiments, are essentially smooth but contain a small number of highfrequency episodes such as spikes. In principle it is possible to employ a different amount of smoothing at di erent spatial loca ..."
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Cited by 4 (1 self)
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Many of the signals to which wavelet methods are applied, including those encountered in simulation experiments, are essentially smooth but contain a small number of highfrequency episodes such as spikes. In principle it is possible to employ a different amount of smoothing at di erent spatial locations, but in the context of wavelets this is so awkward to implement that it is not really practicable. Instead, it is attractive to select the primary resolution level (or smoothing parameter) so as to give good performance for smooth parts of the signal. While this is readily accomplished using a crossvalidation argument, it is unclear whether it has a deleterious impact on performance at highfrequency episodes. In this paper we show that it does not. We derive upper and lower bounds to pointwise rates of convergence for functions whose "spikiness" increases with sample size. (This allows us to model contexts where wavelet methods have to work hard to recover highfrequency events.) We show that, in order to achieve optimal rates of convergence, it is necessary for the primary resolution level of the empirical wavelet transform to vary with location, sometimes extensively. Nevertheless, the convergence rate penalty incurred through using a nonvarying resolution level, chosen to provide good performance for coarsescale features, equals a factor that is less than the logarithm of sample size.
ADAPTATION TO HIGH SPATIAL INHOMOGENEITY BASED ON WAVELETS AND ON LOCAL LINEAR SMOOTHING * JIANQING FAN!
, 1993
"... ABSTRACT. We develop mathematical models for functions with aberrant, highfrequency episodes, and describe the ability of waveletbased estimators to capture those features. Our results have a genuinely local character, in that they describe pointwise asymptotic properties of curve estimators. Prev ..."
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ABSTRACT. We develop mathematical models for functions with aberrant, highfrequency episodes, and describe the ability of waveletbased estimators to capture those features. Our results have a genuinely local character, in that they describe pointwise asymptotic properties of curve estimators. Previous accounts of the performance of wavelet methods have been based on global rates of convergence uniformly over very large function classes; in marked contrast, we establish local convergence rates for single function. We allow those functions to depend on sample size, so that we may describe the extent to which sample size influences the type of short, sharp aberrations that may be accurately recovered from noisy observations. It is shown that wavelet methods based on thresholding, and employing a relatively arbitrary level of primary resolution, capture highfrequency episodes with an accuracy that is within a logarithmic factor of being optimal. We point out that this factor derives from the estimators being somewhat oversmoothed, with systematic errors of larger order than their stochastic errors. That undersmoothing is, in turn, a consequence of inadequate choice of primary resolution. In principle, this difficulty may be overcome by adjusting the primary resolution level in an adaptive way, but that is not a practically appealing proposition, not least because of its computational complexity. By way of contrast, methods based on more traditional smoothing approaches can be applied locally to obtain estimators that outperform wavelet methods in terms of pointwise convergence rates. In particular, we show that estimators based on local linear smoothing attain the optimal convergence rates, even in the presence of unusually high frequencies in the curve. Moreover, these local smoothing methods are straightforward, in both conception and execution.