连续对称下的最优化和采样：实例和李群理论
Optimization and Sampling Under Continuous Symmetry: Examples and Lie Theory
摘要：
在过去的几年里，对称性的概念提供了一个强大而重要的镜头来观察理论计算机科学、统计学、机器学习、量子推理和隐私等领域中出现的几个优化或采样问题。在这里，我们提出了两个例子的非凸优化和采样的连续对称性发挥-隐式或显式-一个关键的作用，在开发高效的算法。这些例子依赖于非凸对称流形和凸多面体之间的深层和隐藏的联系，并且具有很强的可推广性。为了制定和理解这些概括，我们介绍了李群理论-一个不可或缺的数学工具包，用于捕获和处理连续对称性。我们首先介绍李群，李代数的基础知识，以及与它们相关的伴随作用，我们还提到了李代数的分类定理。随后，我们提出了Kostant的凸性定理，并展示了它如何使我们能够减少线性优化问题的轨道上的李群线性优化问题的多面体。最后，我们提出了Harish-Chandra和Harish-Chandra-Itzykson-Zuber（HCIZ）公式，将李群上的配分函数（积分）转换为相应（离散）Weyl群上的和，从而实现有效的采样算法。
英文摘要：
In the last few years, the notion of symmetry has provided a powerful and essential lens to view several optimization or sampling problems that arise in areas such as theoretical computer science, statistics, machine learning, quantum inference, and privacy. Here, we present two examples of nonconvex problems in optimization and sampling where continuous symmetries play -- implicitly or explicitly -- a key role in the development of efficient algorithms. These examples rely on deep and hidden connections between nonconvex symmetric manifolds and convex polytopes, and are heavily generalizable. To formulate and understand these generalizations, we then present an introduction to Lie theory -- an indispensable mathematical toolkit for capturing and working with continuous symmetries. We first present the basics of Lie groups, Lie algebras, and the adjoint actions associated with them, and we also mention the classification theorem for Lie algebras. Subsequently, we present Kostant's convexity theorem and show how it allows us to reduce linear optimization problems over orbits of Lie groups to linear optimization problems over polytopes. Finally, we present the Harish-Chandra and the Harish-Chandra--Itzykson--Zuber (HCIZ) formulas, which convert partition functions (integrals) over Lie groups into sums over the corresponding (discrete) Weyl groups, enabling efficient sampling algorithms.
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