Functional Regression and Classification for Data Supported on Complex Geometries

该项目旨在为位于复杂几何形状上的功能/成像数据开发新的统计方法。由于成像技术的进步，这些数据现在在医学，生物学和气候学等领域无处不在。也许最常见的任务是使用这些复杂的数据（例如，皮层表面的大脑活动）通过回归和分类模型来预测响应变量（例如，年龄或疾病状态）。该项目将开发一个新的回归和分类框架，利用包括偏微分方程和微分几何在内的数学工具，整合现有的额外信息，并定义更准确的模型。此外，还将提供有效的计算实现和对模型在以前看不见的数据上的性能的理论保证。新模型的软件实现将公开提供。研究活动将为下一代统计学家和数据科学家的跨学科研究培训提供众多机会。正在开发的框架将把当前的功能数据方法推广到现代成像数据分析所产生的复杂环境。具体而言，研究活动包括开发正则化线性模型和广义线性模型，用于多维非线性域上支持的函数数据的预测。这些模型将被制定为一个无限维的最小化空间的光滑函数。为了有效地近似他们的解决方案，该项目将采用偏微分方程和变分法的数值分析工具。此外，新的框架将被推广到的情况下，功能预测显示张量结构，可以利用它来控制解决方案的复杂性，通过low-rank constraints.This award reflects NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

This project aims to develop novel statistical methods for functional/imaging data that are located on complex geometries. Thanks to advances in imaging technology, such data are now ubiquitous in the fields of medicine, biology, and climatology, among others. Perhaps the most common task is to use these complex data (for example, brain activity on the cortical surface) to predict a response variable (for example, age or disease status) by employing regression and classification models. The project will develop a novel framework for regression and classification that leverages mathematical tools including partial differential equations and differential geometry to integrate additional information available and define more accurate models. Moreover, efficient computational implementations and theoretical guarantees on the models' performance on previously unseen data will also be provided. The software implementations of the new models will be made publicly available. The research activities will offer numerous opportunities for interdisciplinary research training of the next generation of statisticians and data scientists.The framework under development will generalize current functional data methodology to complex settings that arise from the analysis of modern imaging data. Specifically, the research activities include development of regularized linear models and generalized linear models for predictors that are functional data supported on multidimensional non-linear domains. These models will be formulated as an infinite-dimensional minimization over spaces of smooth functions. To efficiently approximate their solutions, the project will employ tools from numerical analysis of partial differential equations and elements of calculus of variations. Further, the new framework will be generalized to situations where the functional predictors display tensor structure, which can be leveraged to control the complexity of the solution via low-rank constraints.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.