A Theoretical Framework for Discovering Groups and Unitary Representations via Tensor Factorization

TL;DR

This work introduces HyperCube, an operator-valued tensor factorization framework that intrinsically biases toward discovering groups and their unitary representations. By decomposing the objective into a base term and a misalignment term, the authors isolate a collinear manifold where group-like algebraic structures emerge, and prove a unitary-collinear factorization corresponds to the regular representation of a group. They formulate a Collinearity Dominance Conjecture to connect local collinearity results to global landscape behavior, providing conditional proofs that global minima for group isotopes are unitary and that non-group operations incur higher costs. Empirical experiments across Latin-square structures support the dominance hypothesis and demonstrate that HyperCube’s objective can serve as a differentiable proxy for associativity and isotopy structure. The findings suggest a novel inductive bias toward full-rank, unitary representations, with potential to enhance symmetry discovery and out-of-distribution generalization in AI systems.

Abstract

We analyze the HyperCube model, an \textit{operator-valued} tensor factorization architecture that discovers group structures and their unitary representations. We provide a rigorous theoretical explanation for this inductive bias by decomposing its objective into a term regulating factor scales ($\mathcal{B}$) and a term enforcing directional alignment ($\mathcal{R} \geq 0$). This decomposition isolates the \textit{collinear manifold} ($\mathcal{R}=0$), to which numerical optimization consistently converges for group isotopes. We prove that this manifold admits feasible solutions exclusively for group isotopes, and that within it, $\mathcal{B}$ exerts a variational pressure toward unitarity. To bridge the gap to the global landscape, we formulate a \textit{Collinearity Dominance Conjecture}, supported by empirical observations. Conditional on this dominance, we prove two key results: (1) the global minimum is achieved by the unitary regular representation for groups, and (2) non-group operations incur a strictly higher objective value, formally quantifying the model's inductive bias toward the associative structure of groups (up to isotopy).

Figures (1)

• Figure 1: Empirical verification of Strong Collinearity Dominance. Scatter plots showing the normalized objective terms ($\tilde{\mathcal{H}}, \tilde{\mathcal{R}}, \tilde{\mathcal{B}}$) versus the normalized associativity violation $\tilde{n}_v$ for Latin squares corresponding to loops of orders $n \in \{5, 6, 7, 8\}$. (Middle) The misalignment penalty $\tilde{\mathcal{R}}$ grows linearly with non-associativity ($c_R \approx 0.50$). (Right) The base term $\tilde{\mathcal{B}}$ decreases linearly ($c_B \approx 0.14$), confirming variational trade-off. (Left) The total objective $\tilde{\mathcal{H}}$ exhibits a net positive slope ($c_H \approx 0.36$), confirming that the misalignment penalty dominates the trade-off, forcing the global minimum to the group structure ($\tilde{n}_v = 0$).

Theorems & Definitions (35)

• lemma 1: Base Term $\mathcal{B}$
• definition 1: Misalignment Matrices
• lemma 2: Decomposition of $\mathcal{H}$
• proof
• lemma 3: Regular Representation Certificate
• proof
• lemma 4: Synchronization
• proof
• lemma 5: Homomorphism and Equivalence to Regular Representation
• proof