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_i=f(t_i )+σz_i, i=0, …, n-1,t_i=i/n, where the z_i are independent and identically distributed standard Gaussian random variables. The reconstruction fˆ*_n is defined in the wavelet domain by translating all the empirical wavelet coefficients of d toward 0 by an amount σ·√(2log (n)/n). The authors prove two results about this type of estimator. [Smooth]: with high probability fˆ*_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