Efficient computation of mean-magnetization signs in the SK model

Construct an efficient algorithm for computing the signs of the mean magnetization vector, sign(E_{π_{β,h}}[σ_i]) for all coordinates, in the high-field Sherrington–Kirkpatrick model, so that the oracle-centered sampling result can be implemented without assuming access to these signs.

Background

The oracle-centered sampler improves the required external-field threshold to within a 1 + o(1) factor of the Almeida–Thouless and weak Almeida–Thouless thresholds, but it assumes that the signs of the mean magnetization vector are provided as input.

The signs determine the appropriate spin-centering around which the disagreement set is sparse. The paper does not provide a method for obtaining them efficiently, so the result is conditional on this additional oracle information.

References

We remark that Theorem~\ref{thm:oracle_centered_sampling} is a conditional result that requires access to $\star$, the signs of the mean magnetization vector; we leave the efficient computation of $\star$ as an important open problem.

— High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression  (2609.08873 - Kumar et al., 8 Sep 2026) in Appendix, Section “Sampling around the mean,” final paragraph after Theorem (oracle-centered sampling)