Inspired by constraints from physical law, equivariant machine learning restricts the learning to a hypothesis class where all the functions are equivariant with respect to some group action. Irreducible representations or invariant theory are typically used to parameterize the space of such functions. In this article, we introduce the topic and explain a couple of methods to explicitly parameterize equivariant functions that are being used in machine learning applications. In particular, we explicate a general procedure, attributed to Malgrange, to express all polynomial maps between linear spaces that are equivariant under the action of a group $G$, given a characterization of the invariant polynomials on a bigger space. The method also parametrizes smooth equivariant maps in the case that $G$ is a compact Lie group.

翻译：受物理定律约束的启发，等变机器学习将学习过程限制在假设类中，其中所有函数均关于某个群作用具有等变性。不可约表示或不变性理论通常被用于参数化此类函数的空间。本文介绍了这一主题，并阐释了机器学习应用中用于显式参数化等变函数的几种方法。特别地，我们阐述了一种归功于马尔格朗日的通用方法，该方法通过刻画更大空间上的不变多项式，能够表达线性空间之间所有关于群 $G$ 作用等变的多项式映射。当 $G$ 为紧致李群时，该方法同样能参数化光滑等变映射。