CAREER: High-dimensional Tensor Learning: The Good, the Bad, and the Pragmatic

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。高阶张量数据集在现代数据科学应用中无处不在。张量提供了经典低阶方法无法捕获的数据结构的有效表示。然而，经验上的成功揭示了无数的新挑战。与矩阵不同，高阶张量问题在计算上很困难，其统计性质关键取决于算法的选择。PI计划研究一系列张量问题的基本计算与统计权衡。PI将为高维张量估计开发一套统计学习理论、高效算法和数据驱动解决方案。开发的工具将允许领域科学家检查复杂的张量数据，从而为传统分析无法解决的问题提供解决方案。该计划将为不同背景的学员建立一个开放、共融的教育环境，以扩大他们对STEM的参与。教育和研究将通过开发新课程、提供暑期研究培训机会以及让未被充分代表的学生参与研究来整合。该项目将集中在三个主要研究领域：(i)具有统计和计算最优性的参数张量模型；（ii）高阶张量的非参数估计与补全；（iii）具有结构约束的预测张量神经网络模型。PI将研究大范围结构张量的固有低维性，包括但不限于低秩、非负性、块结构和平滑性。优化景观将研究涉及指数族张量、正交可分解张量、矩量方法和深度张量神经网络的非凸问题。新的框架将填补统计预言和解决高阶高维张量问题的经验算法之间的空白。该研究将应用于各种数据问题，如脑连接数据的分类、推荐系统中的模式检测和组学数据集成。将发布带有详细文件的软件包，以促进可重复的研究。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Higher-order tensor datasets are rising ubiquitously in modern data science applications. A tensor provides an effective representation of a data structure that classical low-order methods fail to capture. However, empirical success has uncovered a myriad of new challenges. Unlike matrices, higher-order tensor problems are computationally hard, and the statistical properties crucially hinge on the choice of algorithms. The PI plans to investigate the fundamental computational-statistical tradeoffs for a range of tensor problems. The PI will develop a suite of statistical learning theory, efficient algorithms, and data-driven solutions for high-dimensional tensor estimation. The developed tools will allow domain scientists to examine complex tensor data, thereby providing solutions to questions that traditional analyses cannot address. The PI will broaden participation in STEM by establishing an open, inclusive education environment for trainees with diverse backgrounds. Education and research will be integrated through developing new courses, providing summer research-training opportunities, and engaging underrepresented students in the research. The project will focus on three major research areas: (i) parametric tensor models with statistical and computational optimality; (ii) nonparametric estimation and completion for high-rank tensors; (iii) predictive tensor neural network models with structure constraints. The PI will investigate the intrinsic low-dimensionality for a wide range of structured tensors, including, but not limited to, low-rankness, non-negativity, block-structure, and smoothness. Optimization landscape will be studied for non-convex problems involving exponential-family tensors, orthogonal decomposable tensors, methods-of-moment tensors, and deep tensor neural networks. The new framework will fill in the gap between statistical oracles and the empirical algorithms for addressing higher-order high-dimensional tensor problems. The research will be applied to a variety of data problems, such as classification of brain connectivity data, pattern detection in recommendation systems, and omics data integration. Software packages will be released with detailed documentations to facilitate reproducible research.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.