CAN LOCAL ENERGY GEOMETRY PREDICT PER-PATTERN RETRIEVAL RELIABILITY IN DENSE ASSOCIATIVE MEMORIES?

Capacity analyses of dense associative memories (DAMs) characterize global
phase transitions but cannot predict which individual patterns will fail retrieval
in a given finite-size system. We propose the basin isolation metric Iµ(σ), a
Hessian-free diagnostic that measures the anharmonicity of the energy landscape
around each stored pattern by probing radial energy profiles along random tan-
gent directions. Evaluating on a spherical DAM with cubic interactions (n=3)
across N ∈ {100, 200, 500, 1000} in the near-transition regime, we find that at
N ≤ 200, Iµ outperforms pairwise overlap baselines (AUC-ROC up to 0.68),
is reasonably robust to its scale parameter, and captures nonlinear geometric in-
formation not fully captured by simple overlap statistics. However, with a fixed
number of probing directions K, the diagnostic degrades at N ≥ 500, consistent
with random tangent sampling becoming increasingly sparse relative to the grow-
ing tangent-space dimensionality. These results provide a geometric perspective
on per-pattern retrieval variability and clarify the regime where local landscape
probing remains informative.