De-noising by soft-thresholding

Donoho and Johnstone (1994) proposed a method for reconstructing an unknown function f on [0,1] from noisy data d/sub i/=f(t/sub i/)+/spl sigma/z/sub i/, i=0, ..., n-1,t/sub i/=i/n, where the z/sub i/ are independent and identically distributed standard Gaussian random variables. The reconstruction f/spl circ/*/sub n/ is defined in the wavelet domain by translating all the empirical wavelet coefficients of d toward 0 by an amount /spl sigma//spl middot//spl radic/(2log (n)/n). The authors prove two results about this type of estimator. [Smooth]: with high probability f/spl circ/*/sub n/ is at least as smooth as f, in any of a wide variety of smoothness measures. [Adapt]: the estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. The present proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model. >

[1]  T. W. Anderson.  The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities , 1955 .

[2]  G. Wahba,et al.  A completely automatic french curve: fitting spline functions by cross validation , 1975 .

[3]  Charles A. Micchelli,et al.  A Survey of Optimal Recovery , 1977 .

[4]  Derick Wood,et al.  Completeness of Context-Free Grammar Forms , 1981, J. Comput. Syst. Sci..

[5]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[6]  C. J. Stone.  OPTIMAL GLOBAL RATES OF CONVERGENCE FOR NONPARAMETRIC ESTIMATORS , 1982 .

[7]  M. R. Leadbetter,et al.  Extremes and Related Properties of Random Sequences and Processes: Springer Series in Statistics , 1983 .

[8]  B. W. Silverman,et al.  Probability, Statistics and Analysis: Some properties of a test for multimodality based on kernel density estimates , 1983 .

[9]  Ker-Chau Li,et al.  From Stein's Unbiased Risk Estimates to the Method of Generalized Cross Validation , 1985 .

[10]  M. Nussbaum.  Spline Smoothing in Regression Models and Asymptotic Efficiency in $L_2$ , 1985 .

[11]  S. Geer.  A New Approach to Least-Squares Estimation, with Applications , 1986 .

[12]  Y. Meyer,et al.  Ondelettes et bases hilbertiennes. , 1986 .

[13]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[14]  I. Daubechies.  Orthonormal bases of compactly supported wavelets , 1988 .

[15]  Gilles Deslauriers,et al.  Symmetric iterative interpolation processes , 1989 .

[16]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[17]  O. Lepskii.  Asymptotic Minimax Estimation with Prescribed Properties , 1990 .

[18]  D. Donoho,et al.  Minimax Risk Over Hyperrectangles, and Implications , 1990 .

[19]  Y. Meyer.  Ondelettes sur l'intervalle. , 1991 .

[20]  G. Weiss,et al.  Littlewood-Paley Theory and the Study of Function Spaces , 1991 .

[21]  Dennis M. Healy,et al.  Noise Reduction By Constrained Reconstructions In The Wavelet-transform Domain , 1991, Proceedings of the Seventh Workshop on Multidimensional Signal Processing.

[22]  David L. Donoho,et al.  Interpolating Wavelet Transforms , 1992 .

[23]  G. Kerkyacharian,et al.  Density estimation in Besov spaces , 1992 .

[24]  Stéphane Mallat,et al.  Singularity detection and processing with wavelets , 1992, IEEE Trans. Inf. Theory.

[25]  I. Johnstone,et al.  Empirical functionals and e cient smoothing parameter selection , 1992 .

[26]  Edward H. Adelson,et al.  Shiftable multiscale transforms , 1992, IEEE Trans. Inf. Theory.

[27]  Iain M. Johnstone,et al.  Estimation d'une densité de probabilité par méthode d'ondelettes , 1992 .

[28]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[29]  D. Donoho.  Smooth Wavelet Decompositions with Blocky Coefficient Kernels , 1993 .

[30]  D. Donoho.  Unconditional Bases Are Optimal Bases for Data Compression and for Statistical Estimation , 1993 .

[31]  I. Daubechies,et al.  Multiresolution analysis, wavelets and fast algorithms on an interval , 1993 .

[32]  D. L. Donoho,et al.  Ideal spacial adaptation via wavelet shrinkage , 1994 .

[33]  D. Donoho.  Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery , 1994 .

[34]  D. Donoho.  Statistical Estimation and Optimal Recovery , 1994 .

[35]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[36]  I. Johnstone,et al.  Wavelet Shrinkage: Asymptopia? , 1995 .

[37]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .