New Statistical Challenges Posed by Multiscale and Adaptive Representations

项目周期：07/01/00--06/30/05该项目将涉及以下具体主题：1.断层成像中的估计：根据初步计算，曲线紧框架表示似乎比传统的断层成像方法实现更快的收敛速度。时频分析中的估计和检验：通过在时频平面上部署曲波，建立了一种新的线性调频线框。这种新的表示法对于检测和估计存在噪声的线性调频信号具有很好的前景。最近在计算视觉方面的工作提出了一个问题：“表示自然图像的最佳基础是什么？”有人认为，在迅速增长的关于这个主题的文献中，许多经验结果可以使用分析构造的表征：脊线和曲线来解释和改进。几何驱动的扩散在应用图像处理中很受欢迎，但其定量性能还没有被很好地理解。建议使用多尺度脊线等最新工具发展一种定量统计理论。稀疏性思想的延伸。寻找“白噪声中的边缘”和“白噪声中的子空间”的问题为推广现有稀疏性思想提供了具有挑战性的及时方向。测试稀疏手段。当随机向量可能表现出未知的稀疏度时，提出了一种自适应方法来检验随机向量的均值是否是非零的。大协方差矩阵最高本征值的渐近性。根据随机矩阵理论的最新进展，制定了一个程序来发展大数据矩阵的主成分的统计理论。按照“可重复研究”的原则，这个项目的软件和数字将在公共领域小波实验室系统的未来版本中提供。近年来，小波和时频方法的研究已经扩大，以构建新的表示系统，包括为特定现象定制的系统。例子包括小波包和余弦包，以及最近的一些系统，如小波包、小脊波、小波包、小波包和小波包。同时，在统计分析和相关领域的研究允许数据本身决定它们自己的最佳表示系统的设计。主成分(即Karhunen-Loeved分解)是这种数据自适应表示的最古老的例子；最近提出了许多较新的概念，例如独立成分分析。提出者在这两个领域都很活跃，创造了新的图像和信号表示，并发展了统计理论来支持自适应的信号表示。目前的项目将(A)寻求最近引入曲线带来的两个机会，(B)解决两个活跃的应用研究领域，计算视觉和几何驱动的扩散，以及(C)解决一些被认为是统计决策理论新发展的核心问题。专题(A)可能对断层摄影术、图像和信号处理的应用工作产生影响，以及(C)可能影响主成分在气候和全球变化研究等领域的应用。

NEW STATISTICAL CHALLENGES POSED BY MULTISCALE AND ADAPTIVEREPRESENTATIONS D.L. Donoho & I.M. Johnstone, PI's.Project period: 07/01/00 -- 06/30/05The project will address the following specific topics:1. Estimation in tomography: the curvelet tight frame ofrepresentation seems, according to preliminary calculations, toachieve faster rates of convergence than traditional tomographicmethods.2. Estimation and testing in time-frequency analysis: a new tightframe of chirplets has been built by deploying curvelets in thetime-frequency plane. The new representation has promise for detection andestimation of chirps in the presence of noise.3. Recent work in computational vision asks ``what is the best basisfor representing natural images?'' It is proposed that many empiricalresults in the rapidly growing literature on this topic might beexplained and improved using analytically-constructed representations:ridgelets and curvelets.4. Geometry-driven diffusions are popular in applied image processing-- but their quantitative performance is not well-understood. It isproposed to develop a quantitative statistical theory using recenttools such as multiscale ridgelets.5. Extensions of Sparsity-Based ideas. Problems of finding `edgels inwhite noise' and `subspaces in white noise' offer challenging andtimely directions in which to generalize existing sparsity ideas.6. Testing Sparse Means. An adaptive approach is suggested fortesting if the mean of a random vector is nonzero, when the vectormight exhibit an unknown degree of sparsity.7. Asymptotics of top eigenvalues of large covariance matrices.A program is set out to develop statistical theory for principalcomponents of large data matrices based on recent progress inrandom matrix theory.Consistent with the principle of ``reproducible research'', softwareand figures from this project will be made available in futurereleases of the public domain WaveLab system.In recent years, research in wavelets and time-frequency methods hasbroadened to construct new systems of representation, includingsystems custom-tailored for specific phenomena. Examples includewavelet packets and cosine packets, and very recently, systems likeedgelets, ridgelets, chirplets, warplets, and curvelets. In parallel,research in statistical analysis and cognate fields allows datathemselves to dictate the design of their own optimal systems ofrepresentation. Principal components (i.e. Karhunen-Loevedecomposition) is the oldest example of such data-adaptiverepresentation; many newer ideas have been proposed recently, such asindependent components analysis. The proposers have beenactive in both domains, creating new image and signalrepresentations and developing statistical theory to underpin adaptivesignal representations. The current project will (a) pursue twoopportunities arising from the recent introduction of curvelets, (b)address two active applied research areas, computational vision andgeometry driven diffusions, and (c) attack some issues which areargued to be at the core of new developments in statistical decisiontheory. Topic (a) may have implications for applied work intomography, image and signal processing, and (c) may impact applieduses of principal components in domains such as climate and globalchange studies.