Sharpen retrieval certificates for correlated overcomplete patterns

Derive sharper retrieval and contraction certificates than the current sigma_max^2(X)-based bounds, especially for correlated overcomplete pattern matrices with N > d, where the existing certificate becomes excessively conservative.

Background

The local retrieval certificate depends linearly on sigma_max2(X). Numerical experiments show that this dependence leads to large gaps between certified and empirical retrieval in correlated and overcomplete regimes, even though the underlying memory may retrieve patterns reliably. The conclusions identify improving this certificate as an explicit remaining problem.

References

Four problems remain open. The first is quantitative rather than qualitative; above the escape energy our results localize the possible discrepancies between continuous and discrete basins but do not bound them, and a measure-theoretic or Hausdorff estimate of the deformation as a function of $\theta$ and $\beta$ would complete the picture; the unstable fixed points recently shown to be attached to faces of the pattern polytope \citep{beise2026} are the natural organizing objects. The second is the combination problem of Section~\ref{sec:sav}; discrete-gradient methods dissipate the true energy exactly at second order but are fully implicit, the SAV scheme is linearly implicit at one softmax evaluation per step but dissipates only a modified energy, and no scheme known to us achieves second order, exact dissipation of $E$ for every step size, a monotone interpolant, and one evaluation per step. The third is sharpening the certificate itself, whose constant $\sigma_{\max}2(X)$ degrades exactly in the correlated overcomplete regime where Hopfield layers operate.

— Basin-Preserving Discretizations of Modern Hopfield Retrieval Dynamics: Energy Cells, Dissipation, and the Attention Limit  (2608.21304 - Villatoro, 21 Aug 2026) in Section 7, Conclusions