在顺应对称约束下的全局最优性
Global optimality under amenable symmetry constraints
AI总结：
本文研究凸函数在顺应群不变下的不变极小值存在性，通过分析最小闭凸不变子集，提出统一工具，并应用于不变核均值嵌入与风险最优耦合，揭示求和技巧与Hunt-Stein定理的联系。
摘要：
考虑一个在变换群作用下不变的凸函数。如果它有极小值点，那么它是否也存在一个不变的极小值点？这个问题的变体出现在非参数统计以及许多相邻领域中。答案取决于函数的选择，以及人们可以粗略地称为问题几何性质的因素——即凸性、群和底层向量空间（通常是无限维的）之间的相互作用。我们观察到，这种几何性质完全编码在空间的最小闭凸不变子集中，并进而研究这些集合，针对顺应但不一定紧致的群。然后我们将这一工具应用于不变最优性问题。它产生了关于不变核均值嵌入和风险最优不变耦合的新结果，并阐明了看似不同的概念之间的关系，例如机器学习中用于构造等变神经网络的求和技巧和统计学中的经典Hunt-Stein定理。
英文摘要：
Consider a convex function that is invariant under an group of transformations. If it has a minimizer, does it also have an invariant minimizer? Variants of this problem appear in nonparametric statistics and in a number of adjacent fields. The answer depends on the choice of function, and on what one may loosely call the geometry of the problem -- the interplay between convexity, the group, and the underlying vector space, which is typically infinite-dimensional. We observe that this geometry is completely encoded in the smallest closed convex invariant subsets of the space, and proceed to study these sets, for groups that are amenable but not necessarily compact. We then apply this toolkit to the invariant optimality problem. It yields new results on invariant kernel mean embeddings and risk-optimal invariant couplings, and clarifies relations between seemingly distinct ideas, such as the summation trick used in machine learning to construct equivariant neural networks and the classic Hunt-Stein theorem of statistics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
