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Group Cross-Correlations with Faintly Constrained Filters

Benedikt Fluhr

TL;DR

This work broadens the scope of $G$-equivariant transformations by replacing the stringent bi-equivariance requirement with weaker stabilizer-invariance constraints on a point-dependent filter $\omega$, enabling cross-correlations on actions with non-compact stabilizers and non-transitive dynamics. It develops a comprehensive framework connecting cross-correlations on bundle sections with orbitwise integral transforms via kernels $\kappa$, and introduces Mackey sections as a flexible bridge between bundle-valued and function-valued representations. The paper provides precise continuity and compatibility conditions, and both directions of translation between filters and kernels are established through explicit constructions involving measures on $G$, its orbits, and stabilizers. The results extend the applicability of $G$-equivariant signal processing and pave the way for neural-network architectures and integral-transform pipelines on general homogeneous spaces beyond the classical compact-stabilizer/unimodular/transitive setting.

Abstract

We provide a notion of group cross-correlations, where the associated filter is not as tightly constrained as in the previous literature. This resolves an incompatibility previous constraints have for group actions with non-compact stabilizers. Moreover, we generalize previous results to group actions that are not necessarily transitive, and we weaken the common assumption of unimodularity.

Group Cross-Correlations with Faintly Constrained Filters

TL;DR

This work broadens the scope of -equivariant transformations by replacing the stringent bi-equivariance requirement with weaker stabilizer-invariance constraints on a point-dependent filter , enabling cross-correlations on actions with non-compact stabilizers and non-transitive dynamics. It develops a comprehensive framework connecting cross-correlations on bundle sections with orbitwise integral transforms via kernels , and introduces Mackey sections as a flexible bridge between bundle-valued and function-valued representations. The paper provides precise continuity and compatibility conditions, and both directions of translation between filters and kernels are established through explicit constructions involving measures on , its orbits, and stabilizers. The results extend the applicability of -equivariant signal processing and pave the way for neural-network architectures and integral-transform pipelines on general homogeneous spaces beyond the classical compact-stabilizer/unimodular/transitive setting.

Abstract

We provide a notion of group cross-correlations, where the associated filter is not as tightly constrained as in the previous literature. This resolves an incompatibility previous constraints have for group actions with non-compact stabilizers. Moreover, we generalize previous results to group actions that are not necessarily transitive, and we weaken the common assumption of unimodularity.
Paper Structure (22 sections, 17 theorems, 155 equations, 1 figure)

This paper contains 22 sections, 17 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Group Cross-Correlations
  3. Generalization to Non-Abelian Groups
  4. Continuity of Cross-Correlations
  5. Constraints as Stabilizer Invariances
  6. Generalization to Vector-Valued Functions
  7. Mackey Sections
  8. The Action on Mackey Sections
  9. Cross-Correlations with a Filter
  10. Orbitwise Integral Transforms
  11. Equivariance of Orbitwise Integral Transforms
  12. Integral Transforms as Cross-Correlations
  13. Example with Real Numbers and Integers
  14. Special Support of Kernel
  15. Comparison to "Bi-Equivariant Kernels"
  16. ...and 7 more sections

Key Result

Lemma 2.2

The map mapping a section ${f \in \Gamma(E)}$ to its associated Mackey section ${\tilde{f} \in M(E)}$ is $G$-equivariant.

Figures (1)

  • Figure 1: The support of $\kappa$ shaded in red.

Theorems & Definitions (38)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 28 more