On the geometric and Riemannian structure of the spaces of group equivariant non-expansive operators
This work addresses foundational challenges in topological data analysis and deep learning by providing a geometric framework for optimizing group equivariant operators, though it appears incremental in extending existing operator theory.
The paper tackles the problem of analyzing geometric properties of spaces of group equivariant non-expansive operators, showing that such spaces can be structured as Riemannian manifolds, enabling gradient descent for cost minimization, and applies this to select representative operators.
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.