Neural-collapse characterization for the graph unconstrained feature model

Consider the graph unconstrained feature model (gUFM) in the objective defined by equation (

) with $K=1$. For each class $c$, let $s_{cc',i}$ denote the fraction of neighbors of node $v_{c,i}$ that belong to class $c'$, and let condition **C** be

(s_{c1,1},\ldots,s_{cC,1})=\cdots=(s_{c1,n},\ldots,s_{cC,n}),\qquad \forall c\in[C].

$$**Neural-collapse characterization conjecture.** The minimizers of the gUFM are collapsed, satisfying equation ($$

), if and only if the graph $\mathcal{G}$ satisfies condition C.

The preceding theorem proves the forward implication under condition C, and proves the converse when either $\sqrt{\lambda_H\lambda_{W_2}}=0$ or $\sqrt{\lambda_H\lambda_{W_2}}>0$ with $G$ regular. The conjecture asserts that the regularity or symmetry restriction can be removed; the status beyond the theorem's hypotheses is not established here.