Orbit recovery from invariants of low degree in representations of finite groups
Dan Edidin, Josh Katz
TL;DR
The paper addresses orbit recovery in representations of finite groups using low-degree invariants. It develops a constructive Jennrich-type approach showing that, for the regular representation over any infinite field, invariants of degree at most $3$ separate generic orbits by exploiting the second and third moment/invariant tensors $T^G_2$ and $T^G_3$, and contractions $T_a,T_b$ to recover the orbit up to a scale. The work extends to subregular representations, detailing how the dihedral and symmetric groups yield obstructions: some subregular representations require degree-$4$ invariants (e.g. in the complete multiplicity-free dihedral representation) and, for the symmetric group, conjectures a sharp threshold $(d^3+6d^2+11d)/6\ge nd$ for list-resolving generic orbits in $\mathbb{C}^{n\times d}$. These results have implications for designing low-degree, invariant-based components in equivariant neural networks and for MRA/cryo-EM, providing concrete algebraic criteria and conjectures for when generic orbit recovery is feasible with minimal degree invariants.
Abstract
Motivated by applications to equivariant neural networks and cryo-electron microscopy we consider the problem of recovering the generic orbit in a representation of a finite group from invariants of low degree. The main result proved here is that invariants of degree at most three separate generic orbits in the regular representation of a finite group defined over any infinite field. This answers a question posed in a 2023 ACHA paper of Bandeira et. al. We also discuss this problem for subregular representations of the dihedral and symmetric groups.