Group equivariant non-expansive operators have been recently proposed as basic components in topological data analysis and deep learning. In this paper we study some geometric properties of the spaces of group equivariant operators and show how a space $\mathcal{F}$ of group equivariant non-expansive operators can be endowed with the structure of a Riemannian manifold, so making available the use of gradient descent methods for the minimization of cost functions on $\mathcal{F}$. As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant non-expansive operators in the considered manifold.

翻译：群等变非扩张算子近期被提出作为拓扑数据分析与深度学习中的基本组件。本文研究了群等变算子空间的一些几何性质，并展示了群等变非扩张算子空间 $\mathcal{F}$ 如何被赋予黎曼流形结构，从而使梯度下降法能够用于 $\mathcal{F}$ 上的代价函数最小化。作为该方法的一个应用，我们还描述了一种在所考虑的流形中选取有限组代表性群等变非扩张算子的过程。