Sharp Capacity Thresholds in Linear Associative Memory: From Winner-Take-All to Listwise Retrieval

2026-05-06

AI Summary

This study investigates the theoretical storage capacity limits of linear associative memory models under distinct retrieval criteria. Focusing on Gaussian-distributed linear memory matrices, it distinguishes between winner-take-all (top-1) and list-based retrieval mechanisms, leveraging tools from random matrix theory, extreme value statistics, and convex optimization to establish precise asymptotic capacity characterizations. The work demonstrates that top-1 retrieval incurs a logarithmic capacity penalty and proves that correlated matrix constructions achieve optimal logarithmic scaling in this regime. Furthermore, it introduces the tail-average margin (TAM) criterion, which enables quadratic capacity scaling under list-based retrieval, yielding closed-form expressions for the critical load and the distribution of retrieval scores. The analysis also leads to the conjecture that the top-1 threshold satisfies \(d^2 \sim 2n \log n\).

Abstract

How many key-value associations can a $d\times d$ linear memory store? We show that the answer depends not only on the $d^2$ degrees of freedom in the memory matrix, but also on the retrieval criterion. In an isotropic Gaussian model for the stored pairs, we show that top-1 retrieval, where every signal must beat its largest distractor, requires the logarithmic model-size scale $d^2\asymp n\log n$. We prove that the correlation matrix memory construction, which stores associations by superposing key-target outer products, achieves this scale through a sharp phase transition, and that the same scaling is necessary for any linear memory. Thus the logarithm is the intrinsic extreme-value price of winner-take-all decoding. We next consider listwise retrieval, where the correct target need not be the unique top-scoring item but should remain among the strongest candidates. To formalize this regime, we propose the Tail-Average Margin (TAM), a convex upper-tail criterion that certifies inclusion of the correct target in a controlled candidate list. Under this listwise retrieval criterion, the capacity follows the quadratic scale $d^2\asymp n$. At load $n/d^2\toα$, we develop an exact asymptotic theory for the TAM empirical-risk minimizer through a two-parameter scalar variational principle. The theory has a rich phenomenology: in the ridgeless limit it yields a closed-form critical load separating satisfiable and unsatisfiable phases, and it predicts the limiting laws of true scores, competitor scores, margins, and percentile profiles. Finally, a small-tail extrapolation further leads to the conjectural sharp top-1 threshold $d^2\sim 2n\log n$.

Problem

• associative memory
• capacity threshold
• linear memory
• retrieval criterion
• extreme-value statistics

Innovation

• linear associative memory
• capacity threshold
• listwise retrieval
• Tail-Average Margin
• phase transition