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

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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 functionestimation, 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 shrinkagemethods which include the standard term-by-term thresholding as aspecial case. Block thresholding is studied via the approach ofideal 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 estimatorswith good empirical performance, all of which may be useful in differentestimation 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 adaptabilityand information-pooling in general orthogonal series estimation, ofwhich wavelets are a special case. Preliminary results show thatseparable rules lack adaptability; they are necessarily not fully rate-adaptive. A key to adaptively achieve the exact minimax rate isinformation-pooling. I will further carry out research in this topicand will derive a lower bound on the amount of information-poolingrequired 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 forthe construction of fully adaptive estimators. Besides theoreticalinvestigation, 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.