Recovery of Functions via Moments: Hausdorff Case

通过矩恢复功能：Hausdorff 案例

• 批准号: 0906639
• 负责人: Robert Mnatsakanov
• 金额: $ 12万
• 依托单位: West Virginia University Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906639&HistoricalAwards=false
• 关键词: Recovery; Functions; via; Moments; Hausdorff

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The intellectual merit of this proposal is connected to the problem of recovering a multivariate function from its assigned moments and the problem of density estimation for high-dimensional data. The problem of recovering a function from its moments is a special case of the classical moment problem, which concentrates mainly on the questions of the existence and uniqueness of a function with specified moments. The importance of the probabilistic moment problem (Hamburger, Stieltjes, and Hausdorff) can be explained by its application in many statistical inverse problems. For example, in tomography, the moments of an object (a function) are uniquely defined by the x-rays (projections) of the object being imaged. Many inversion formulas are derived by inverting the moment generating function and the Laplace transform. However, there are only a few approaches for recovering functions via moments. This can be explained by the unstable behavior of the current methods (e.g., the Maximum Entropy method applied even in the one-dimensional case) when the higher order moments are involved. The investigator develops a new approach, which yields a stable procedure for recovering functions within the context of the multivariate Hausdorff moment problem. Apart from being an alternative to the traditional estimation technique, this approach is applicable in situations where other methods can not be applied. For example, one cannot use a traditional method, e.g., kernel smoothing, when the observed data are the moments. The results obtained within this project will have broad impacts not only in the multidimensional Hausdorff moment problem, in the theory of non-parametric estimation in indirect models (deconvolution and demixing), and in entropy estimation of high-dimensional macromolecules, but also in numerous applications in areas of critical importance, such as image analysis, computed tomography, molecular physics, and homeland security. In particular, in computed tomography, when only a few projections are available, the problem of image reconstruction becomes ill-posed, and hence, perfect reconstruction is impossible. The investigator shows that proposed approach provides a uniform approximation rate, which is an important issue in approximation theory. Besides, in many statistical inverse problems, e.g., those based on convolutions, mixtures, multiplicative censoring, and right-censoring, the moments of the unobserved distribution of actual interest can be easily estimated from the transformed moments of the observed distributions. In all such models, one can recover a function analytically from its moments by means of proposed technique. In the area of homeland security, the iris classification problem represents another field, where moment-recovered constructions will have an impact.

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。这一建议的智力价值是连接到从其分配的时刻和高维数据的密度估计问题的多变量函数的恢复问题。从矩恢复函数的问题是经典矩问题的一个特例，经典矩问题主要研究具有特定矩的函数的存在唯一性问题。概率矩问题（Hamburger，Stieltjes，and Hausdorff）的重要性可以通过它在许多统计逆问题中的应用来解释。例如，在断层摄影中，物体的矩（函数）由被成像物体的X射线（投影）唯一定义。通过对矩母函数的反演和拉普拉斯变换，导出了许多反演公式。然而，只有少数几种方法可以通过矩来恢复函数。这可以通过当前方法的不稳定行为来解释（例如，即使在一维情况下也应用最大熵方法）。研究人员开发了一种新的方法，这产生了一个稳定的程序，恢复功能的背景下，多变量Hausdorff矩问题。除了作为传统估计技术的替代方案之外，这种方法还适用于其他方法无法应用的情况。例如，不能使用传统方法，例如，核平滑，当观测数据是矩时。 该项目中获得的结果不仅将在多维豪斯多夫矩问题、间接模型中的非参数估计理论（去卷积和去混合）以及高维大分子的熵估计中产生广泛影响，而且还将在图像分析、计算机断层扫描、分子物理和国土安全等至关重要领域的众多应用中产生广泛影响。特别是，在计算机断层摄影中，当只有少数投影可用时，图像重建的问题变得不适定，因此，完美的重建是不可能的。研究表明，所提出的方法提供了一个统一的近似率，这是一个重要的问题，在近似理论。此外，在许多统计逆问题中，例如，基于卷积、混合、乘性删失和右删失的那些，实际感兴趣的未观测分布的矩可以容易地从观测分布的变换矩估计。在所有这样的模型中，人们可以通过所提出的技术从其时刻解析地恢复函数。在国土安全领域，虹膜分类问题代表了另一个领域，其中力矩恢复结构将产生影响。