Enumerate all optimal coupling configurations for larger thermodynamic-learning networks

Determine all coupling configurations in the set \(\{j_<\}\) that minimize the total training energy for larger all-to-all thermodynamic-learning networks, including networks beginning with the \(4\!-4\!-4\) architecture, together with their associated degeneracies.

Background

The thermodynamic-learning model trains slow dichotomous couplings $\{j\}$ by minimizing an effective energy over the training input patterns. In the prior-limit regime $\bm\beta\to\bm\beta_0^{prior}$, the relevant configurations $\{j_<\}$ are those that realize the global optimization over all training inputs. Their enumeration is needed to compute the complete distribution of memorization fidelities and degeneracies rather than results based only on a detected subset.

For sufficiently small architectures, the authors determine these configurations by exact enumeration. As the network grows, however, the coupling-configuration space becomes extremely large, making the optimal configurations difficult to locate. The unresolved task is therefore to find the complete set $\{j_<\}$, particularly for architectures at and beyond the $4\!-4\!-4$ case, where the reported numerical results use only a restricted sample.