Recovery of Functions via Moments: Hausdorff Case

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。该建议的智力价值与从其指定矩中恢复多元函数的问题和高维数据的密度估计问题有关。函数矩恢复问题是经典矩问题的一种特殊情况，主要研究具有特定矩的函数的存在性和唯一性问题。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，在断层扫描中，一个物体的矩（函数）是由被成像物体的x射线（投影）唯一定义的。许多反演公式是通过对矩生成函数和拉普拉斯变换进行反求得到的。然而，通过矩恢复函数的方法很少。这可以用当前方法（例如，即使在一维情况下应用的最大熵方法）在涉及高阶矩时的不稳定行为来解释。研究者开发了一种新的方法，该方法在多元豪斯多夫矩问题的背景下产生了恢复函数的稳定过程。除了作为传统评估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，当观测到的数据是矩时，不能使用传统的方法，例如核平滑。该项目获得的结果将不仅在多维Hausdorff矩问题、间接模型（反卷积和除混）的非参数估计理论和高维大分子的熵估计方面产生广泛的影响，而且在图像分析、计算机断层扫描、分子物理和国土安全等至关重要的领域也有许多应用。特别是在计算机断层扫描中，当只有少量的投影可用时，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，该方法提供了均匀的近似速率，这是近似理论中的一个重要问题。此外，在许多统计反问题中，例如基于卷积、混合、乘法审查和右审查的统计反问题，可以很容易地从观察分布的变换矩中估计实际感兴趣的未观察分布的矩。在所有这样的模型中，人们可以通过所提出的技术从函数的矩中解析地恢复函数。在国土安全领域，虹膜分类问题代表了另一个领域，其中矩恢复结构将产生影响。

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.