Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields

• URL: http://arxiv.org/abs/2603.08758v1
• Date: Sun, 08 Mar 2026 10:41:23 GMT
• Title: Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields
• Abstract summary: 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.<n>We apply this characterization to Equivariant Neural Fields, extending them to arbitrary group actions and homogeneous conditioning spaces.
• Score: 43.58107597764876
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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