Block Thresholding Methods for Adaptive Wavelet Function Estimation: Theory and Applications

Purdue Research Foundation, West Lafayette IN

Investigators

Abstract

This research studies two interrelated function estimation problems, nonparametric regression and linear inverse problems, using wavelet methods via the approach of block thresholding and ideal adaptation with oracle. The goals are to build a bridge between the traditional multivariate normal decision theory and the adaptive wavelet function estimation, and to develop a family of estimators that achieve simultaneously three objectives: adaptivity, spatial adaptivity, and computational efficiency. A major innovation and a consistent theme throughout the research is the use of block shrinkage methods which include the standard term-by-term thresholding as a special case. Block thresholding is studied via the approach of ideal adaptation with oracle. It will be demonstrated that block thresholding serves as a bridge between the classical normal decision theory and adaptive wavelet function estimation. This leads to a systematic way of developing a coherent set of rate-optimal estimators with good empirical performance, all of which may be useful in different estimation problems. To fully understand why block thresholding works ``better'' than the standard term-by-term thresholding, and more generally, separable rules, I will explore the connection between adaptability and information-pooling in general orthogonal series estimation, of which wavelets are a special case. Preliminary results show that separable rules lack adaptability; they are necessarily not fully rate-adaptive. A key to adaptively achieve the exact minimax rate is information-pooling. I will further carry out research in this topic and will derive a lower bound on the amount of information-pooling required for achieving full global adaptivity. These results together will offer a deeper understanding of the benefit of information-pooling in nonparametric function estimation, and also serve as a guide for the construction of fully adaptive estimators. Besides theoretical investigation, I am also interested in applications of the wavelet methods. I am collaborating with colleagues on using wavelet methods for archiving and retrieval of medical images from tomographic databases.