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.

翻译：考虑一个在某个变换群作用下不变的凸函数。如果该函数存在极小值点，那么它是否也存在一个不变的极小值点？该问题的变体出现在非参数统计及多个相关领域中。答案取决于函数的选择，以及可以粗略地称为问题几何结构的因素——即凸性、群结构及通常为无限维的底层向量空间之间的相互作用。我们观察到，这种几何结构完全编码于空间中最小的闭凸不变子集中，并进而研究这些集合，针对的是可迁但未必紧致的群。随后，我们将这一工具集应用于不变最优性问题。由此获得了关于不变核均值嵌入和风险最优不变耦合的新结果，并澄清了若干看似不同概念之间的联系，例如机器学习中用于构建等变神经网络的求和技巧，以及统计学中经典的亨特-斯坦定理。