Theoretical Analysis of HyperCube Objective for Group Representation Learning

The HyperCube model is a promising tensor factorization framework for discovering latent group structures in data. A foundational conjecture posits that its global minima correspond to unitary group representations, but a proof has remained elusive. We make significant theoretical progress by decomposing the HyperCube objective into a base term dependent on matrix norms and a misalignment term. We introduce the Perfect Alignment Conjecture, which states that this misalignment must vanish at any stationary point for the optimization to capture a true group. Under this condition, we prove that all local minima are in fact global and are unitarily equivalent to the group's regular representation, thus conditionally resolving the original conjecture. Our analysis reveals HyperCube’s unique inductive bias for full-rank, unitary solutions, distinguishing it from the typical low-rank bias in deep learning models.