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，这种数据现在在医学、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，大脑皮质表面的活动)通过使用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式和微分几何在内的数学工具，整合现有的更多信息，并确定更准确的模型。此外，还将为模型在以前未见过的数据上的性能提供有效的计算实现和理论保证。新型号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大量机会。正在开发的框架将把目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中。具体地说，研究活动包括开发用于预报器的正则化线性模型和广义线性模型，这些模型是多维非线性域上支持的函数数据。这些模型将被表示为光滑函数空间上的无限维极小化。为了有效地近似它们的解，该项目将使用偏微分方程组和变分原理的数值分析工具。此外，新的框架将被推广到功能预测器显示张量结构的情况，可以利用张量结构通过低级约束来控制解决方案的复杂性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.