Collaborative Research: AF: Small: Phase Transitions in Sampling Related Problems

图形模型是一种广泛使用的工具，可以简洁地表示高维数据，理解物理和生物系统的基本构件和相互作用。这些模型已经出现在各种科学领域。在物理学中，它们被用来理解铁磁材料的热力学性质，并且在研究物理系统中的相变时是不可或缺的。在生物学中，这些模型是利用系统发育模型中的遗传数据来推断进化史的基本工具。在贝叶斯推理等计算任务的机器学习中，图形模型是普遍存在的。这个项目解决了基本的计算任务，这些任务对于研究、构建和利用图形模型至关重要。该项目将让本科生参与社会科学背景下图形模型的研究。学习图形模型有两个核心任务：学习和采样。学习问题的重点是从系统的宏观行为推断底层图形模型的内部结构。相反，相关采样问题的目标是有效地模拟学习或推断的图形模型的热力学行为。这个项目将开发新的算法，并更广泛地了解采样和学习的计算复杂性以及几个相关问题。这个项目的一个共同主题是将这些采样和推理相关问题的计算复杂性与统计-物理相变联系起来。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Graphical models are a widely used tool to succinctly represent high-dimensional data and to understand the fundamental building blocks and interactions of physical and biological systems. These models have appeared in a variety of scientific fields. In physics they are used to understand the thermodynamic properties of ferromagnetic materials and are integral in the study of phase transitions in physical systems. In biology these models are a fundamental tool for inferring evolutionary history using genetic data in phylogenetic models. Graphical models are ubiquitous in machine learning for computational tasks such as Bayesian inference. This project addresses fundamental computational tasks that are crucial for studying, constructing, and utilizing graphical models. The project will involve undergraduate students in research involving graphical models in social science settings.There are two core tasks for studying graphical models: learning and sampling. The learning problem is focused on inferring the inner structure of the underlying graphical model from the macroscopic behavior of the system. In contrast, the goal of the associated sampling problem is to efficiently simulate the thermodynamic behavior of a learned or inferred graphical model. This project will develop new algorithms, and more generally understand the computational complexity of sampling and learning as well as several related problems. A common theme in this project is connecting the computational complexity of these sampling- and inference-related problems with statistical-physics phase transitions.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.