Hyperbolic wavelet thresholding methods and the curse of dimensionality through the maxiset approach

Abstract

In this paper we compute the maxisets of some denoising methods (estimators) for multidimensional signals based on thresholding coefficients in hyperbolic wavelet bases. That is, we determine the largest functional space over which the risk of these estimators converges at a chosen rate. In the unidimensional setting, refining the choice of the coefficients that are subject to thresholding by pooling information from geometric structures in the coefficient domain (e.g., vertical blocks) is known to provide ‘large maxisets’. In the multidimensional setting, the situation is less straightforward. In a sense these estimators are much more exposed to the curse of dimensionality. However we identify cases where information pooling has a clear benefit. In particular, we identify some general structural constraints that can be related to compound functional models and to a minimal level of anisotropy.