Quasi-linear wavelet estimation

The main paradigm of the modern wavelet theory of spatial adaptation formul ated by Donoho and Johnstone is that there is a divergence between the line ar minimax adaptation theory and the heuristic guiding algorithm developmen t that leads to the necessity of using strongly nonlinear adaptive threshol ded methods. On the other hand, it is well known that linear adaptive estim ates 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 asympt otic properties of strongly nonlinear procedures like the benchmark SureShr ink? The answer is "yes," and we discuss quasi-linear estimation both theor etically and via a Monte Carlo study. In particular, I show that, asymptoti cally, a quasi-linear procedure not only matches properties of SureShrink o ver the Besov scale, but also allows us to relax familiar assumptions and s olve a long standing problem of rate and sharp optimal estimation of monoto ne functions. For the case of small sample sizes and functions that contain spiky/jumps parts and smooth parts, a quasi-linear estimate performs excep tionally well in terms of visual aesthetic appeal, approximation, and data compression.