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

本文利用小波方法，通过块阈值法和oracle的理想自适应，研究了两个相互关联的函数估计问题——非参数回归和线性逆问题。目标是在传统的多元正态决策理论和自适应小波函数估计之间建立一座桥梁，并开发一组同时实现三个目标的估计器：适应性、空间适应性和计算效率。在整个研究中，一个主要的创新和一致的主题是使用块收缩方法，其中包括标准的逐项阈值法作为特殊情况。采用oracle的理想自适应方法对块阈值进行了研究。将证明块阈值是经典正态决策理论和自适应小波函数估计之间的桥梁。这导致了一种系统的方法来开发一组具有良好经验性能的连贯的速率最优估计器，所有这些估计器都可能在不同的估计问题中有用。为了充分理解为什么块阈值法比标准逐项阈值法“更好”，以及更一般地说，可分离规则，我将探索一般正交序列估计中适应性和信息池之间的联系，其中小波是一个特殊情况。初步结果表明：可分离规则缺乏适应性；它们不一定是完全适应速率的。自适应实现精确的极大极小率的关键是信息池。我将进一步开展这一主题的研究，并将得出实现完全全局适应性所需的信息池量的下界。这些结果一起将提供对信息池在非参数函数估计中的好处的更深入的理解，并且也作为构建完全自适应估计器的指南。除了理论研究之外，我对小波方法的应用也很感兴趣。我正在与同事合作，使用小波方法从层析数据库中归档和检索医学图像。

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.