Collaborative Research: New Regression Models and Methods for Studying Multiple Categorical Responses

在许多科学研究领域，包括生物工程、流行病学、基因组学和神经科学，一个重要的任务是建立多个分类结果和大量预测因子之间的关系模型。例如，在癌症研究中，根据数千个基因的表达，对患者是否患有a、B或C亚型癌症以及死亡风险的高低进行建模是至关重要的。然而，现有的统计方法要么无法应用，要么无法捕捉响应变量之间的复杂关系，要么导致难以解释的模型，从而产生很少的科学见解。pi通过开发多种新的统计方法来解决这一缺陷。对于每种新方法，pi将提供理论证明和快速计算算法。与研究生和本科生一起，pi还将创建公开可用的软件，使学术界和工业界的应用程序成为可能。本项目旨在解决多元分类数据分析中的一个基本问题：如何在给定一组常见的高维预测因子的情况下，简洁地对许多分类随机变量的联合概率质量函数进行建模。pi将通过使用张量分解、降维以及凸和非凸优化等新兴技术来解决这个问题。本课题主要研究三个方向：(1)条件概率张量低秩分解的潜变量方法；(2)一种新的多元广义线性回归框架内禀降维的重叠凸惩罚；(3)利用Tucker张量分解上的显式秩约束，提出一种基于直接非凸优化的低秩张量回归方法。不同于对预测因子单独回归每个（单变量）分类响应的方法，新的模型和方法将允许从业者描述响应之间复杂且经常有趣的依赖关系。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。

In many areas of scientific study including bioengineering, epidemiology, genomics, and neuroscience, an important task is to model the relationship between multiple categorical outcomes and a large number of predictors. In cancer research, for example, it is crucial to model whether a patient has cancer of subtype A, B, or C and high or low mortality risk given the expression of thousands of genes. However, existing statistical methods either cannot be applied, fail to capture the complex relationships between the response variables, or lead to models that are difficult to interpret and thus, yield little scientific insight. The PIs address this deficiency by developing multiple new statistical methods. For each new method, the PIs will provide theoretical justifications and fast computational algorithms. Along with graduate and undergraduate students, the PIs will also create publicly available software that will enable applications across both academia and industry.This project aims to address a fundamental problem in multivariate categorical data analysis: how to parsimoniously model the joint probability mass function of many categorical random variables given a common set of high-dimensional predictors. The PIs will tackle this problem by using emerging technologies on tensor decompositions, dimension reduction, and both convex and non-convex optimization. The project focuses on three research directions: (1) a latent variable approach for the low-rank decomposition of a conditional probability tensor; (2) a new overlapping convex penalty for intrinsic dimension reduction in a multivariate generalized linear regression framework; and (3) a direct non-convex optimization-based approach for low-rank tensor regression utilizing explicit rank constraints on the Tucker tensor decomposition. Unlike the approach of regressing each (univariate) categorical response on the predictors separately, the new models and methods will allow practitioners to characterize the complex and often interesting dependencies between the responses.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.