CAREER: Deformations in statistics, cosmology and image analysis

光滑的可逆变换或变形正在迅速成为现代数据分析中的重要工具。变形的非线性性质使这些物体极其强大，同时也使它们具有估计和理论探索的挑战性。这项建议致力于发展和从理论上理解应用于三个具体研究领域的变形：统计学、宇宙学和图像分析。通过与Stein方法的惊人联系，分析了用于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，研究人员重点介绍了最优传输理论中的最新结果，该理论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的欧拉-拉格朗日特征，这可以显著地减轻与估计相关的典型计算负担。一个结果是使这些方法可供统计从业者在一系列广泛的问题中广泛使用：非参数和半参数密度估计、估计宇宙学中的引力透镜和后验抽样技术，仅举几例。这一建议的另一个学术价值是两个新提出的宇宙微波背景弱透镜变形估计的科学分支：一个是小波/Slepian二次估计，另一个是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的最成功的工具之一。对透镜的精确估计很重要，原因有很多，包括但不限于了解宇宙结构、约束宇宙学参数和探测重力波。这位研究人员建议使用小波和Slepian多锥度来调整二次估计，以适应局部前景污染物和天空切割，这是大多数现代宇宙测量中普遍存在的特征。研究人员提出了一种新的贝叶斯估计器，它有可能极大地改变引力透镜研究的完成方式以及它们与其他天文测量的集成方式。光滑的可逆变换或变形正迅速成为现代数据分析中的重要工具。它们在计算解剖学领域获得了惊人的成功，在计算解剖学领域，产生变形的时变矢量场流被用来统计分析医学fMRI图像并量化异常的形态结构。在宇宙学中，形变被用来模拟暗物质密度涨落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融入现代科学方法，但估计变形的统计特性在很大程度上还没有得到探索。这项建议致力于发展和从理论上理解应用于三个具体研究领域的变形：统计学、宇宙学和图像分析。该项目产生的工具将是有用的，不仅在统计学上，而且在从遗传学到机器学习的其他科学和技术分支中也是有用的。例如，在物理学领域，引力透镜估计产生的潜在科学进展可能对科学理解和科学研究的未来产生广泛影响。此外，培训研究生和本科生掌握成功驾驭跨学科工作所需的工具，并为独立研究做好准备，变得越来越重要。该提案的跨学科性质将培养一种合作文化，这种文化将触及统计教育的基本原理，并将加深与统计学和其他物理科学的联系。此外，通过研究和教育的融合，该提案将教授研究生和本科生研究技能。结果将是两个方面。首先，它将培养研究生成为创造性的独立研究人员，他们可以在学术环境中做出贡献。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.