Non-Convex Landscapes and High-Dimensional Latent Variable Models

非凸景观和高维潜变量模型

• 批准号: 1916198
• 负责人: Zhou Fan
• 金额: $ 18.27万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2022-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1916198&HistoricalAwards=false
• 关键词: Non; Convex; Landscapes; Dimensional; Latent

In many fields of science and engineering, probabilistic latent variable models are a powerful and widely-used tool for drawing inferences from complex data. They provide a flexible framework by modeling the complexity in observed data as arising from interactions between simpler random and unobserved quantities. Latent variable models used in modern applications are often high-dimensional, and this leads to both statistical and computational challenges for inference: Surprising phenomena emerge in which structure in one latent variable can create spurious and problematic artifacts in classical inference procedures for another. These classical procedures also commonly lead to non-convex optimization problems over a large number of parameters, which are difficult to computationally solve.This research will study a flexible framework by modeling the complexity in observed data as arising from interactions between simpler random and unobserved quantities. The aim is in answering the following questions: How and why can one source of latent variation lead to artifacts in classical statistical estimates for another? What are the geometric properties of objective function landscapes in these models that render them difficult to optimize? And, can we design improved inferential procedures that correct for these artifacts and are easier to compute? The research will apply techniques from random matrix theory, free probability theory, and statistical physics to obtain a better understanding of these questions in high-dimensional settings.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.

在许多科学和工程领域，概率潜变量模型是从复杂数据中得出推论的强大且广泛使用的工具。它们提供了一个灵活的框架，通过建模的复杂性，在观察到的数据所产生的更简单的随机和未观察到的数量之间的相互作用。现代应用中使用的潜变量模型通常是高维的，这导致了推理的统计和计算挑战：出现了令人惊讶的现象，其中一个潜变量的结构可以在经典推理过程中为另一个潜变量创建虚假和有问题的伪影。这些经典的程序也通常会导致非凸优化问题的大量参数，这是很难计算solved.This研究将研究一个灵活的框架，通过建模的复杂性，在观察到的数据之间的相互作用所产生的简单的随机和不可观测的数量。其目的是回答以下问题：如何以及为什么一个潜在的变化源导致文物在经典的统计估计为另一个？在这些模型中，目标函数景观的几何属性是什么，使得它们难以优化？而且，我们是否可以设计改进的推理程序，纠正这些文物，更容易计算？该研究将应用随机矩阵理论、自由概率论和统计物理学的技术，以更好地理解高维环境中的这些问题。该奖项反映了NSF的法定使命，并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，被认为值得支持。

项目成果

Iterative Alpha Expansion for estimating gradient‐sparse signals from linear measurements

• DOI: 10.1111/rssb.12407
• 发表时间: 2019-05
• 期刊: Journal of the Royal Statistical Society: Series B (Statistical Methodology)
• 作者: Sheng Xu;Z. Fan

Spectral Graph Matching and Regularized Quadratic Relaxations: Algorithm and Theory

• 发表时间: 2020-07
• 作者: Z. Fan;Cheng Mao;Yihong Wu;Jiaming Xu

Principal components in linear mixed models with general bulk

• DOI: 10.1214/20-aos2010
• 发表时间: 2019-03
• 期刊: The Annals of Statistics
• 作者: Z. Fan;Yi Sun;Zhichao Wang

Tree-Projected Gradient Descent for Estimating Gradient-Sparse Parameters on Graphs

• 发表时间: 2020-05
• 作者: Sheng Xu;Z. Fan;S. Negahban

Surfing: Iterative optimization over incrementally trained deep networks

• 发表时间: 2019-07
• 期刊: ArXiv
• 作者: Ganlin Song;Z. Fan;J. Lafferty