Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields

arXiv (Cornell University)

Abstract

Many geometric learning problems require invariants on heterogeneous product spaces, i.e., products of distinct spaces carrying different group actions, where standard techniques do not directly apply. We show that, when a group $G$ acts transitively on a space $M$, any $G$-invariant function on a product space $X \times M$ can be reduced to an invariant of the isotropy subgroup $H$ of $M$ acting on $X$ alone. Our approach establishes an explicit orbit equivalence $(X \times M)/G \cong X/H$, yielding a principled reduction that preserves expressivity. We apply this characterization to Equivariant Neural Fields, extending them to arbitrary group actions and homogeneous conditioning spaces, and thereby removing the major structural constraints imposed by existing methods.

Authors 5

1. Alejandro García-Castellanos

   University of Amsterdam

   Affiliation as printed

   Amsterdam Machine Learning Lab (AMLab) , University of Amsterdam

2. Gijs Bellaard

   Eindhoven University of Technology

   Affiliation as printed

   Department of Mathematics and Computer Science , Eindhoven University of Technology

3. Remco Duits

   Eindhoven University of Technology

   Affiliation as printed

   Department of Mathematics and Computer Science , Eindhoven University of Technology

4. Daniël M. Pelt Aachen Leiden Institute of Advanced Computer Science

   Leiden University

   Affiliation as printed

   Leiden Institute of Advanced Computer Science , Universiteit Leiden

5. Erik J. Bekkers

   University of Amsterdam

   Affiliation as printed

   Amsterdam Machine Learning Lab (AMLab) , University of Amsterdam