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Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields

Alejandro García-Castellanos, Gijs Bellaard, Remco Duits, Daniel Pelt, Erik J Bekkers

TL;DR

This work shows 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, yielding a principled reduction that preserves expressivity.

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.

Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields

TL;DR

This work shows that, when a group acts transitively on a space , any G-invariant function on a product space can be reduced to an invariant of the isotropy subgroup of acting on X alone, yielding a principled reduction that preserves expressivity.

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 acts transitively on a space , any -invariant function on a product space can be reduced to an invariant of the isotropy subgroup of acting on alone. Our approach establishes an explicit orbit equivalence , 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.
Paper Structure (29 sections, 9 theorems, 31 equations, 1 algorithm)

This paper contains 29 sections, 9 theorems, 31 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Result
  3. Orbit Equivalence
  4. Generalized Reduction to the Isotropy
  5. Properties of the Reduction
  6. Flexibility in Choice of Reduction
  7. The Group as a Homogeneous Space
  8. Use Case: Equivariant Neural Fields
  9. Discussion and Future Work
  10. Related Work
  11. Mathematical Preliminaries
  12. Reduction to the isotropy
  13. Proof of Orbit equivalence
  14. Properties of Reduction to the isotropy
  15. Proof of Flexibility of Choice of Reduction
  16. ...and 14 more sections

Key Result

Lemma 2.1

Let $G$ act transitively on the set $M$ and (not necessarily transitively) on the set $X$. Fix $p_0 \in M$ and let $H := \mathrm{Stab}_G(p_0)$. Then the map $\Phi : (X \times M)/G \to X/H$, defined by is a bijection between the orbit spaces $(X \times M)/G$ and $X/H$.

Theorems & Definitions (20)

  • Lemma 2.1: Orbit Equivalence
  • Remark 2.1: Homogeneous space identification
  • Theorem 2.1: Universal Property of Quotients; see dummit_abstract_2003
  • Theorem 2.2: Generalized Reduction to the Isotropy
  • proof
  • Remark 2.2
  • Corollary 2.1
  • Corollary 2.2
  • Remark 2.3
  • Definition B.1: Orbit Separation; see dym_low_2023
  • ...and 10 more