Can Local Energy Geometry Predict Per-Pattern Retrieval Reliability in Dense Associative Memories?

Tatiana Petrova

Keywords: energy landscape, basin of attraction, per-pattern retrieval, anharmonicity, high-dimensional geometry, dense associative memory, Hopfield network

TL;DR: We propose a Hessian-free basin isolation metric that predicts per-pattern retrieval failure in dense associative memories at moderate dimensions, but identify a fundamental dimensional barrier where all local diagnostics degrade.

Abstract:

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 $I_\mu(\sigma)$, a Hessian-free diagnostic that measures the anharmonicity of the energy landscape around each stored pattern by probing radial energy profiles along random tangent directions.
Evaluating on a spherical DAM with cubic interactions ($n{=}3$) across $N \in \{100, 200, 500, 1000\}$ in the near-transition regime, we find that at $N \leq 200$, $I_\mu$ outperforms pairwise overlap baselines (AUC-ROC up to $0.68$), is reasonably robust to its scale parameter, and captures nonlinear geometric information not fully captured by simple overlap statistics.
However, with a fixed number of probing directions $K$, the diagnostic degrades at $N \geq 500$, consistent with random tangent sampling becoming increasingly sparse relative to the growing tangent-space dimensionality.
These results provide a geometric perspective on per-pattern retrieval variability and clarify the regime where local landscape probing remains informative.

Submission Number: 16