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

本研究利用小波方法，以Oracle为工具，透过区块门槛与理想适应的方法，研究两个相互关联的函数估计问题：非参数回归与线性逆问题。我们的目标是建立一个传统的多元正态决策理论和自适应小波函数估计之间的桥梁，并开发一个家庭的估计，同时实现三个目标：自适应性，空间自适应性和计算效率。一个主要的创新和一致的主题，在整个研究是使用块收缩方法，其中包括标准的长期长期阈值作为特例。利用oracle的自适应方法研究了块阈值分割。它将被证明，块阈值作为经典的正常决策理论和自适应小波函数估计之间的桥梁。 这导致了一个系统的方式开发一套连贯的率最优estimatorswith良好的经验性能，所有这些都可能是有用的差分testimation问题。 为了充分理解为什么块阈值比标准的逐项阈值，更普遍的是，可分离的规则“更好”，我将探讨adaptability和信息池之间的连接一般正交序列估计，其中小波是一个特例。初步结果表明，可分离的规则缺乏适应性，他们不一定是完全的速率自适应。信息池是自适应地获得精确的极大极小速率的关键。我将进一步开展研究在这个topicand将得出一个下界的信息poolingrequired实现全面的全球adaptability。这些结果将提供一个更深层次的理解信息池在非参数函数估计的好处，也作为一个指导forthe建设完全自适应估计。除了理论研究外，我对小波方法的应用也很感兴趣。我正在与同事合作，使用小波方法从断层扫描数据库中存档和检索医学图像。

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.