Mathematical Sciences: Adaptive Estimation: New Tools, New Settings

提案：DMS 95-05151 PI(s)：Iain Johnstone - 1981，David Donoho - 1984 机构：斯坦福大学 标题：自适应估计、新工具、新设置 摘要：该研究开发了理论统计中的基本工具，以帮助理解此类自适应过程，并展示了如何调整它们，使它们能够识别噪声且稳定。该方法的基础是a）预言机的思想，它非常清楚如何理想地适应表示，b）在存在噪声数据的情况下适应的目标是量化可实现的过程（没有关于对象的特权信息）可以模仿预言机的程度，以及c）尽可能接近预言机的过程设计。 该项目还通过比较不同类型的预言机（例如时间频率预言机和时间尺度预言机）来开发比较不同适应方案的方法。 这是我们早期关于小波的结果的产物，该方法用于表明小波具有近乎理想的空间自适应特性。 此外，正在开发用于实施和系统测试此类方法的计算环境。 作为提案者早期小波工作的进一步发展，该项目研究了小波收缩的许多改进和扩展，例如在分类、置信带、相关数据和正交基选择方面。 本研究旨在在表示和分析信号、图像和其他对象的自适应方法的一般领域中开发统计理论和计算工具。 一方面，这个项目是由信号处理、图像处理、时频分析、语音处理分析领域的人们对适应性的极高兴趣推动的。 这些领域几乎每天都会出现新的信号表示形式，以及选择表示形式的新原则。 这种表示对于数据压缩很有用，但这不是这里的主要兴趣；但它们对于噪声消除和信号解释也很有用，这是统计学家需要考虑的重要领域。 另一方面，该项目的目标是发展统计理论，使人们能够清楚地理解这种自适应方案。信号处理中提出的一些最具适应性的方案对传统统计思维提出了真正的挑战。 例如，一些自适应方案通过数千或数百万个信号表示进行搜索（显式或隐式）以获得最终结果。 统计学家在考虑将此类方法应用于噪声信号时，通过训练被迫询问结果在多大程度上仅反映了通过噪声窥探的效果，检测实际上由于噪声和积极搜索而产生的伪结构。 该研究旨在开发理论统计中的基本工具，以帮助理解这种自适应程序，并展示如何调整它们，使它们能够识别噪声并保持稳定。 该项目可能有两个副产品。 首先，一些结果可能对信号、图像、语音和时间/频率方面的“自适应程序发明者”社区以及相关社区是刺激和/或有用的。 其次，理论工作可能会激发统计学家更有兴趣在这些方向上做出进一步的贡献。

Proposal: DMS 95- 05151 PI(s): Iain Johnstone - 1981, David Donoho - 1984 Institution: Stanford Title: Adaptive Estimation, New Tools, New Settings Abstract: The research develops fundamental tools in theoretical statistics to aid understanding of such adaptive procedures, and shows how to tune them so they are noise-cognizant and stable. Underlying the approach are a) the idea of oracles, which know perfectly well how to adapt representations ideally, b) the idea that the goal of adaptation in the presence of noisy data is to quantify how closely realizable procedures (which do not have privileged information about the object) can mimic an oracle, and c) the design of procedures coming as close as possible to the oracle. The project also develops methods for comparing different adaptation schemes by comparing oracles of different kinds, for example time-frequency oracles and time-scale oracles. This is an outgrowth of our earlier results on wavelets, where this approach was used to show that wavelets have a property of being nearly-ideally spatially adaptive. In addition a computational environment is being developed for implementing and systematically testing such approaches. As a further outgrowth of the proposers' earlier work on wavelets, the project studies a number of improvements and extensions of wavelet shrinkage, for example in the directions of classification, confidence bands, correlated data and selection of orthogonal bases. This research seeks to develop statistical theory and computational tools in the general area of adaptive methods of representing and analyzing signals, images and other objects. On the one hand, this project is prompted by the extremely high interest in adaptation on the part of people working in fields of signal processing, image processing, time-frequency analysis, speech processing analysis. New signal representations appear in these fields almost daily, along with new principles fo r selecting representations. Such representations are useful for data compression, which is not the main interest here; but they are also useful for noise removal and signal interpretation, which are important areas for statisticians to consider. On the other hand, this project is prompted by the goal of developing statistical theory which can give clear understanding of such adaptive schemes. Some of the most highly adaptive schemes being suggested in signal processing pose a real challenge to traditional statistical thinking. For example, some adaptation schemes search (either explicitly or implicitly) through thousands or millions of representations of a signal in order to arrive at their final result. A statistician, when thinking about applying such methods to a noisy signal, is by training forced to ask to what extent the result merely reflects the effects of snooping through noise, detecting pseudo-structure which actually due to noise, and due to the vigorous search. The research seeks to develop fundamental tools in theoretical statistics to aid understanding of such adaptive procedures, and shows how to tune them so they are noise-cognizant and stable. This project may have two spin-offs. First, some of the results may be stimulating and/or useful to the community of ``inventors of adaptive procedures'' in signal, image, speech, and time/frequency, and related communities. Second, the theoretical work may stimulate statisticians to take more interest in making further contributions in such directions.