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.
