On the Behavior of MDL Denoising

We consider wavelet denoising based on minimum description length (MDL) principle. The derivation of an MDL denoising criterion proposed by Rissanen involves a renormalization whose effect on the resulting method has not been well understood so far. By inspecting the behavior of the method we obtain a characterization of its domain of applicability: good performance in the low variance regime but over-fitting in the high variance regime. We also describe unexpected behavior in the theoretical situation where the observed signal is pure noise. An interpretation for the renormalization is given which explains both the empirical and theoretical findings. For practitioners we point out two technical pitfalls and ways to avoid them. Further, we give guidelines for constructing improved MDL denoising methods.