Refine the conditional-energy representation to eliminate the energy-histogram mismatch

Develop a more refined finite-dimensional representation of the WCRG conditional energies for the frustrated soft-spin BNNNI model in order to resolve the mismatch between the microscopic-energy distributions of original Monte Carlo configurations and WCRG-generated configurations.

Background

The WCRG sampler represents each exact conditional energy with a finite-dimensional quadratic-plus-local ansatz. Because this ansatz is not fully expressive, the conditional distributions generated during reconstruction differ from the exact conditional distributions of the microscopic Gibbs measure, even with infinitely many training samples. Consequently, WCRG-generated configurations follow an approximate synthetic hierarchical measure, and the histogram of the original microscopic energy evaluated on those configurations does not match the corresponding Monte Carlo histogram.

The paper identifies a more expressive finite-dimensional representation of the conditional energies as the required remedy, but does not provide such a representation. This unresolved approximation problem is important for improving the accuracy of configuration-wide observables while retaining the computational advantages of learned multiscale sampling.

References

Resolving the mismatch in Fig.~\ref{fig:magn-ene-hist}(b) would require a more refined finite-dimensional representation of $\bar E_j$, but such representation is not immediately available. This issue is therefore left as future work.

— Overcoming critical slowing down in frustrated spin systems by learned multiscale sampling  (2608.31114 - Bandini et al., 31 Aug 2026) in Section 3.3, “Configuration-wide observables,” paragraph “Energy”