Non-Convex Landscapes and High-Dimensional Latent Variable Models

在科学和工程的许多领域中，概率潜变量模型是一种强大而广泛使用的工具，用于从复杂数据中进行推断。它们提供了一个灵活的框架，将观测数据的复杂性建模为由更简单的随机量和未观测量之间的相互作用产生的。现代应用程序中使用的潜在变量模型通常是高维的，这给推理带来了统计和计算方面的挑战：出现了令人惊讶的现象，其中一个潜在变量的结构可能在经典推理过程中为另一个潜在变量创建虚假和有问题的工件。这些经典过程也通常导致大量参数上的非凸优化问题，这些问题难以计算解决。这项研究将研究一个灵活的框架，通过建模观察数据的复杂性，作为更简单的随机和未观察到的数量之间的相互作用。目的是回答以下问题：一个潜在变异的来源如何以及为什么会导致对另一个潜在变异的经典统计估计中的伪影？在这些模型中，目标函数景观的哪些几何特性使得它们难以优化？并且，我们是否可以设计改进的推理过程来纠正这些工件并且更容易计算？本研究将应用随机矩阵理论、自由概率论和统计物理等技术，在高维环境下更好地理解这些问题。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 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.