CAREER: Deformations in statistics, cosmology and image analysis

平滑可逆变换或变形正在迅速成为现代数据分析的重要工具。变形的非线性性质使这些物体非常强大，同时也使它们难以估计和理论上探索。本提案致力于变形的发展和理论理解，应用于三个特定的研究领域：统计学、宇宙学和图像分析。通过与Stein方法的惊人联系，分析了产生非参数和半参数统计估计的估计变形的理论性质。此外，研究者还关注了最近在最优运输理论中发现的结果，这有可能为可变形模板提供严格的理论基础。估计变形的计算方面将利用惩罚最大似然估计的新欧拉-拉格朗日表征，这可以显着减轻与估计相关的典型计算负担。一个结果将是使这些方法被统计实践者广泛应用于各种各样的问题：非参数和半参数密度估计，宇宙学中的引力透镜估计和后验抽样技术，等等。该建议的另一个智力优点是两个新提出的宇宙微波背景（CMB）弱透镜变形估计的科学分支：小波/Slepian二次估计和新的贝叶斯透镜估计。引力透镜研究已经成为探索暗物质本质最成功的工具之一。透镜效应的精确估计很重要，原因有很多，包括但不限于理解宇宙结构、约束宇宙学参数和探测重力波。研究者建议使用小波和Slepian多锥来调整二次估计，以适应局部前景污染物和天空切割，这是大多数现代宇宙学调查中普遍存在的特征。研究者提出了一个新的贝叶斯估计，它有可能极大地改变引力透镜研究的方式，以及如何将它们整合到其他天文调查中。平滑可逆变换或变形正在迅速成为现代数据分析的重要工具。它们已经在计算解剖学领域取得了惊人的成功，其中使用时变矢量场流产生变形来统计分析医学fMRI图像并量化异常形态结构。在宇宙学中，变形被用来模拟暗物质密度波动引起的宇宙微波背景的引力扭曲，并导致对宇宙结构的更深层次的理解。尽管这些重要的工具正在被整合到现代科学方法中，但估计变形的统计特性在很大程度上尚未被探索。本提案致力于变形的发展和理论理解，应用于三个特定的研究领域：统计学、宇宙学和图像分析。这个项目产生的工具不仅在统计学中有用，而且在从遗传学到机器学习的其他科学和技术分支中也很有用。例如，在物理学领域，引力透镜估计所带来的潜在科学进步可能对科学理解和科学研究的未来产生广泛的影响。此外，为研究生和本科生提供必要的工具，使他们能够成功地驾驭跨学科工作，并为独立研究做好准备，这变得越来越重要。该建议的跨学科性质将促进一种合作文化，这种文化将达到统计教育的基础，并将加深与统计学和其他物理科学的联系。此外，通过研究与教育的整合，提案将教授研究生和本科生的研究技能。结果将是双重的。首先，它将培养研究生成为能够在学术环境中做出贡献的有创造力的独立研究人员。其次，它将教育本科生驾驭一个重视创造性独立调查的工作环境。

Smooth invertible transformations, or deformations, are fast becoming important tools in modern data analysis. The nonlinear nature of deformations makes these objects extremely powerful while at the same time making them challenging to estimate and theoretically explore. This proposal is dedicated to the development and theoretical understanding of deformations applied to three specific areas of research: statistics, cosmology and image analysis. The theoretical properties of estimated deformations for generating nonparametric and semiparametric statistical estimates are analyzed through a surprising connection with Stein's method. In addition, the investigator focuses on recent results found in the theory of optimal transport, which has the potential to provide a rigorous theoretical foundation for deformable templates. The computational aspects of estimated deformations will utilize a new Euler-Lagrange characterization of a penalized maximum likelihood estimate, which can significantly relieve the typical computational burden associated with estimation. One consequence will be to make these methods available for widespread use by statistical practitioners in a broad range of problems: nonparametric and semiparametric density estimation, estimating gravitational lensing in cosmology and posterior sampling techniques, to name a few. Another intellectual merit of this proposal are the scientific ramifications of two new proposed deformation estimates of weak lensing of the cosmic microwave background (CMB): a wavelet/Slepian quadratic estimator and a new Bayesian lensing estimator. Gravitational lensing studies have become one of the most successful tools for probing the nature of dark matter. The precise estimation of lensing is important for a number of reasons including, but not limited to, understanding cosmic structure, constraining cosmological parameters and detecting gravity waves. The investigator proposes to uses wavelets and Slepian multi-tapers to adapt the quadratic estimate to local foreground contaminants and sky cuts, which are ubiquitous features in most modern cosmological surveys. The investigator proposes a new Bayesian estimator which has the potential to dramatically change the way gravitational lensing studies are done and how they are integrated within other astronomical surveys.Smooth invertible transformations, or deformations, are fast becoming important tools in modern data analysis. They have been used with spectacular success in the field of computational anatomy where time varying vector field flows which generate deformations are used to statistically analyze medical fMRI images and quantify abnormal morphological structure. In cosmology, deformations are used to model gravitational distortions of the cosmic microwave background from dark matter density fluctuations, and have resulted in a deeper understanding of cosmic structure. Even though these important tools are becoming integrated in modern scientific methods, the statistical properties of estimated deformations have been largely unexplored. This proposal is dedicated to the development and theoretical understanding of deformations applied to three specific areas of research: statistics, cosmology and image analysis. The tools resulting from this project will be useful, not only in statistics, but also in other branches of science and technology ranging from genetics to machine learning. In the field of physics, for example, the potential scientific progress resulting from gravitational lensing estimation could a have broad impact on scientific understanding and the future of scientific research. Moreover, it is becoming increasingly important to train graduate and undergraduate students with the tools necessary to successfully navigate interdisciplinary work, and who are prepared for independent research. The interdisciplinary nature of the proposal will foster a culture of collaboration that will reach the fundamentals of statistical education and will deepen ties with statistics and other physical sciences. In addition, through the integration of research and education, the proposal will teach both graduate and undergraduate students research skills. The result will be two fold. First, it will train graduate students to become creative independent researchers who can contribute within an academic environment. Second, it will educate undergraduates to navigate a work environment which values creative independent investigation.