Quasi-linear wavelet estimation

The main paradigm of the modern wavelet theory of spatial adaptation formulated by Donoho and Johnstone is that there is a divergence between the linear minimax adaptation theory and the heuristic guiding algorithm development that leads to the necessity of using strongly nonlinear adaptive thresholded methods. On the other hand, it is well known that linear adaptive estimates are the best whenever an estimated function is smooth. Is it possible to suggest a quasi-linear wavelet estimate, by adding to a linear adaptive estimate a minimal number of nonlinear terms on finest scales, that offers advantages of linear adaptive estimates and at the same time matches asymptotic properties of strongly nonlinear procedures like the benchmark SureShrink? The answer is "yes," and we discuss quasi-linear estimation both theoretically and via a Monte Carlo study. In particular, I show that, asymptotically, a quasi-linear procedure not only matches properties of SureShrink over the Besov scale, but also allows us to relax familiar assumptions and solve a long standing problem of rate and sharp optimal estimation of monotone functions. For the case of small sample sizes and functions that contain spiky/jumps parts and smooth parts, a quasi-linear estimate performs exceptionally well in terms of visual aesthetic appeal, approximation, and data compression.