Universal approximation and statistical complexity for nonlinear groupoid-steerable networks

Establish universal approximation theorems and statistical sample-complexity estimates for nonlinear groupoid-steerable networks in measured and partially equivariant settings, including criteria for density in spaces of continuous or measurable maps satisfying the relevant filtered transport laws and a precise relation between sample complexity and pair-orbit and stabilizer data.

Background

The paper completely characterizes the finite-dimensional linear layer space through joint-stabilizer intertwiners on pair orbits, and its synthetic experiments suggest that the dimension of the equivariant operator space controls generic identification complexity. However, these results do not establish expressive universality for nonlinear networks built from the proposed groupoid-equivariant layers and pointwise nonlinearities.

The unresolved problem is to determine when such networks are dense in appropriate classes of continuous or measurable maps that satisfy the relevant filtered transport laws, and to develop a statistical theory connecting geometric data—particularly pair orbits and stabilizers—to the number of observations required for reliable identification or learning.

References

The finite layer space is completely characterized, but universal approximation for nonlinear groupoid-steerable networks remains to be established in the present measured and partially equivariant setting. One would like criteria ensuring density in spaces of continuous or measurable maps satisfying the relevant filtered transport laws, together with estimates of sample complexity in terms of pair-orbit and stabilizer data. The synthetic experiment suggests that the dimension of the equivariant operator space governs generic identification complexity; a statistical theory should make this relation precise.

— Theory for groupoid equivariant neural networks: an approach for steerable CNNs on bounded domains  (2609.25987 - Ibort et al., 22 Sep 2026) in Section Conclusions and outlook, subsection “Approximation theory and statistical complexity”