New Statistical Challenges Posed by Multiscale and Adaptive Representations

多尺度和自适应表示带来的新统计挑战

• 批准号: 0072661
• 负责人: Iain Johnstone
• 金额: $ 85万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072661&HistoricalAwards=false
• 关键词: New; Statistical; Challenges; Posed; Multiscale

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.

多尺度和自适应表示带来的新统计挑战 D.L. Donoho 和 I.M. Johnstone，PI's。项目周期：07/01/00 -- 06/30/05该项目将解决以下具体主题：1。层析成像中的估计：根据初步计算，曲波紧表示框架似乎比传统层析成像方法能实现更快的收敛速度。 2．时频分析中的估计和测试：通过在时频平面上部署曲线小波，构建了新的线性调频小帧。 新的表示有望在存在噪声的情况下检测和估计线性调频脉冲。3．计算视觉方面的最新工作提出了“表示自然图像的最佳基础是什么？”有人提出，有关该主题的快速增长的文献中的许多经验结果可以使用分析构造的表示来解释和改进：脊波和曲线。4。几何驱动的扩散在应用图像处理中很流行，但它们的定量性能尚不清楚。建议使用多尺度脊波等最新工具开发定量统计理论。5．基于稀疏性的思想的扩展。寻找“白噪声中的边缘”和“白噪声中的子空间”的问题为推广现有的稀疏性思想提供了具有挑战性且及时的方向。6．测试稀疏均值。 当向量可能表现出未知程度的稀疏性时，建议采用自适应方法来测试随机向量的平均值是否非零。7。大协方差矩阵的顶特征值的渐近。基于随机矩阵理论的最新进展，制定了一个计划来开发大数据矩阵主成分的统计理论。与“可重复研究”的原则相一致，该项目的软件和数据将在公共域 WaveLab 系统的未来版本中提供。近年来， 小波和时频方法已经扩展到构建新的表示系统，包括为特定现象定制的系统。 示例包括小波包和余弦包，以及最近的系统，如edgelet、ridgelet、chirplet、warplet 和curvelet。 与此同时，统计分析和同源领域的研究允许数据本身决定其最佳表示系统的设计。 主成分（即 Karhunen-Loevedecomposition）是此类数据自适应表示的最古老的示例；最近提出了许多更新的想法，例如独立成分分析。 提议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信号表示。 当前的项目将（a）抓住最近引入的曲线小波带来的两个机会，（b）解决两个活跃的应用研究领域，计算视觉和几何驱动扩散，以及（c）解决一些被认为是统计决策理论新发展核心的问题。主题（a）可能对断层扫描、图像和信号处理的应用工作产生影响，（c）可能影响主要成分在气候和全球变化研究等领域的应用。