CAREER: Inference for High-Dimensional Structures via Subspace Learning: Statistics, Computation, and Beyond

高维数组通常产生于现代科学技术研究，并已成为现代统计和数据科学的中心主题。一些领域，如遗传学、微生物组研究、脑成像、高光谱成像等，产生了大量的高维阵列数据；而在其他一些领域，数据可以重铸成高维阵列形式，以便于分析。在这些情况下，目标参数往往是高维/高阶的，但重要的信息可能在于各种结构条件导致的降维子空间。如何有效地利用这些子空间带来了巨大的统计和计算挑战。本项目旨在从子空间学习的角度解决这些挑战。通过考虑降维和低阶子空间，PI旨在通过开发具有统计和计算优势的新方法和理论来解决一系列统计和机器学习问题。该项目将沿着三个主要方向进行：(I)通过重要的子空间草图对高维数组进行快速估计和推断；(Ii)在理论上有保证的高阶聚类；(Iii)通过张量序列参数化进行超高阶张量奇异值分解。这项研究将适用于涉及高维矩阵和张量数据的各种主题，如遗传学和基因组、强化学习、神经成像分析、材料科学、推荐器设计等。PI还将为新算法开发用户友好的软件包，并将其提供给公众使用。PI致力于通过参与研究项目来培训学生，特别是那些来自STEM中代表性不足的群体的学生。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。

High-dimensional arrays commonly arise from modern scientific and technological research and have been a central topic in modern statistics and data science. Some areas such as genetics, microbiome studies, brain imaging, hyperspectral imaging, etc., yield a large amount of high-dimensional array data; while in some other areas, data can be recast into high-dimensional array form to facilitate analysis. In these situations, the target parameter is often high-dimensional/high-order, but the important information may lie in dimension-reduced subspaces induced by various structural conditions. How to efficiently exploit these subspaces poses significant statistical and computational challenges. This project aims to address these challenges from a perspective of subspace learning. By taking into account dimension-reduced and low-order subspaces, the PI aims to address a series of statistical and machine learning questions by developing new methodologies and theories with statistical and computational advantages. This project will progress along three major directions: (i) fast estimation and inference for high-dimensional arrays via important subspace sketching; (ii) high-order clustering with theoretical guarantees; (iii) ultrahigh-order tensor singular value decomposition via a tensor-train parameterization. The research will be applicable to a variety of topics involving high-dimensional matrix and tensor data, such as genetics and genomes, reinforcement learning, neuroimaging analysis, material science, recommender design, etc. The PI will also develop user-friendly software packages for the new algorithms and make them available for public use. The PI is committed to training students, especially those from groups underrepresented in STEM, through involvement in the research project.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.