Extend cell preservation to nonsmooth normalized retrieval models

Extend the energy-cell preservation theorem to nonsmooth normalized retrieval models in which the convex part contains the indicator of a ball, the gradient of the convex part is replaced by a subdifferential, and the corresponding mirror-family update becomes a normalize-after-mix operation.

Background

The smooth Bregman generalization establishes cell preservation for damped difference-of-convex iterations and a certified overrelaxed window under bounded asymmetry. Normalized retrieval models fall outside the stated assumptions because their convex part includes an indicator function and is nonsmooth. Extending the argument to this setting is identified as the most valuable single unresolved extension.

References

The fourth is the nonsmooth side of Section~\ref{sec:generalized}; Corollary~\ref{cor:cells-overrelax} closes the smooth Bregman extension by preserving cells throughout a certified overrelaxed window, but normalized retrieval models place the constraint inside the convex part through the indicator of a ball, so that $\nabla E_{+}$ becomes a subdifferential and the mirror family a normalize-after-mix update. Extending the cell-preservation theorem to that nonsmooth setting appears to us the most valuable single extension left open here.

— 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