On the Performance of Isotropic and Hyperbolic Wavelet Estimators

Abstract

In this paper, we differentiate between isotropic and hyperbolic wavelet bases in the context of multivariate nonparametric function estimation. The study of the latter leads to new phenomena and non trivial extensions of univariate studies. In this context, we fi rst exhibit the limitations of isotropic wavelet estimators by proving that no isotropic estimator is able to guarantee the reconstruction of a function with anisotropy in an optimal or near optimal way. Second, we show that hyperbolic wavelet estimators are well suited to reconstruct anisotropic functions. In particular, for each considered estimator we focus on the rates at which it can reconstruct functions from anisotropic Besov spaces. We then compute the estimator's maxiset, this is the largest functional spaces over which its risk converges at these rates. Our results furnish novel arguments to understand the primordial role of sparsity and thresholding in multivariate contexts, notably by showing the exposure of linear methods to the curse of dimensionality. Moreover, we propose a block thresholding hyperbolic estimator and show its ability to estimate anisotropic functions at the optimal minimax rate and related, the remaining pertinence of information pooling in high dimensional settings.