Spectral Methods for High Dimensional Tensor Data

最近的发展使大规模的多维数据在科学和工程应用中变得容易获得。例子包括多组织、多个体基因表达研究，其中从不同个体的组织收集基因表达谱。另一个例子是DBLP数据库，它被组织成一个作者-词-地点的三向张量，每个条目都表示三元组的同现。尽管张量数据很受欢迎，但使用统计方法来分析高阶张量仍然存在许多挑战。实际上，经典的矩阵谱理论不能直接应用于张量，在最坏的情况下，计算问题变成了NP-困难。因此，分析维数和复杂度不断增加的张量数据需要开发新的统计方法，这是本项目的目标。在本项目中，PI计划开发一个统计模型框架，可扩展算法和相关理论来分析张量值数据。这将使研究人员能够检查张量条目之间和多个张量之间的复杂相互作用，从而为传统矩阵分析无法解决的问题提供解决方案。该项目将集中在三个主要领域：（i）特殊结构或随机张量的谱理论;（ii）从非高斯观测估计低秩张量;（iii）张量值数据的均值和协方差的联合估计。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Recent developments have made large-scale multidimensional data readily available in science and engineering applications. Examples include multi-tissue, multi-individual gene expression studies, in which gene expression profiles are collected from different individuals' tissues. Another example is the DBLP database, which is organized into a three-way tensor of author-by-word-by-venue, and each entry indicates the co-occurrence of the triplets. Despite the popularity of tensor data, there are many challenges to using statistical methods for analyzing higher-order tensors. Indeed, the classical spectral theory for matrices is not directly applicable to tensors, and the computational problem becomes NP-hard in the worst case. Therefore, analyzing tensor data with increasing dimensionality and ever-growing complexity requires the development of novel statistical methods, which is the aim of this project.In this project, the PI plans to develop a framework of statistical models, scalable algorithms, and relevant theories to analyze tensor-valued data. This will allow researchers to examine complex interactions among tensor entries and between multiple tensors, thereby providing solutions to questions that cannot be addressed by traditional matrix analysis. The project will focus on three major areas: (i) spectral theory for specially-structured or random tensors; (ii) estimation of low-rank tensors from non-Gaussian observations; and (iii) joint estimation of mean and covariance for tensor-valued data.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.