Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields

TL;DR

This paper introduces a reduction technique for geometric learning problems involving group actions, enabling more flexible and expressive equivariant neural models by simplifying the invariance conditions.

Contribution

It establishes a general reduction method for $G$-invariant functions on product spaces, extending equivariant neural fields to broader group actions and spaces.

Findings

Reduction of invariants to isotropy subgroup actions

Extension of equivariant neural fields to arbitrary groups

Removal of structural constraints in geometric learning

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.