We ask whether there exists a function or measure that (1) minimizes a given convex functional or risk and (2) satisfies a symmetry property specified by an amenable group of transformations. Examples of such symmetry properties are invariance, equivariance, or quasi-invariance. Our results draw on old ideas of Stein and Le Cam and on approximate group averages that appear in ergodic theorems for amenable groups. A class of convex sets known as orbitopes in convex analysis emerges as crucial, and we establish properties of such orbitopes in nonparametric settings. We also show how a simple device called a cocycle can be used to reduce different forms of symmetry to a single problem. As applications, we obtain results on invariant kernel mean embeddings and a Monge-Kantorovich theorem on optimality of transport plans under symmetry constraints. We also explain connections to the Hunt-Stein theorem on invariant tests.

翻译：我们探讨是否存在一个函数或测度，既满足（1）最小化给定的凸泛函或风险，又（2）满足由可合变换群指定的对称性质。此类对称性质的示例包括不变性、等变性或拟不变性。我们的研究借鉴了Stein和Le Cam的经典思想，以及可合群遍历定理中出现的近似群平均方法。一类在凸分析中被称为轨道体的凸集成为关键要素，我们建立了非参数设定下此类轨道体的性质。我们还展示了如何利用称为上循环的简单工具将不同形式的对称性约化为单一问题。作为应用，我们获得了关于不变核均值嵌入的结果，以及关于对称约束下传输计划最优性的Monge-Kantorovich定理。我们还阐释了与不变检验的Hunt-Stein定理之间的联系。