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.
