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

Forum topic · 小凯 · 2026-05-08

Summary

This arXiv paper (2605.05189) by Nicholas Barnfield, Juno Kim, Eshaan Nichani, Jason D. Lee, and Yue M. Lu studies how many key-value associations a d x d linear memory matrix can store, showing the answer depends on the retrieval criterion, not just the d^2 degrees of freedom. Under an isotropic Gaussian model, top-1 retrieval—where each signal must beat its largest distractor—requires the logarithmic scaling d^2 ~ n log n. The authors prove the correlation matrix memory construction achieves this scale through a sharp phase transition, and that this scaling is necessary for any linear memory, making the logarithm the intrinsic extreme-value price of winner-take-all decoding. For listwise retrieval, they introduce the Tail-Average Margin (TAM), a convex upper-tail criterion, under which capacity follows the quadratic scale d^2 ~ n. An exact asymptotic theory via a two-parameter scalar variational principle yields a closed-form critical load separating satisfiable and unsatisfiable phases in the ridgeless limit, plus limiting laws for scores, margins, and percentile profiles. Small-tail extrapolation conjectures a sharp top-1 threshold d^2 ~ 2n log n.

Paper Overview

Field: Machine Learning Authors: Nicholas Barnfield, Juno Kim, Eshaan Nichani, Jason D. Lee, Yue M. Lu arXiv: 2605.05189

Abstract

How many key-value associations can a d x 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 ~ 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 ~ n. At load n/d^2 -> alpha, 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 ~ 2n log n.

Key Findings

Top-1 retrieval capacity: Scaling d^2 ~ n log n is necessary for any linear memory; the logarithm reflects the extreme-value cost of winner-take-all decoding.
Correlation matrix memory: Achieves this scale with a sharp phase transition.
Listwise retrieval: With the proposed Tail-Average Margin (TAM) criterion, capacity improves to quadratic scaling d^2 ~ n.
Exact asymptotics: A two-parameter scalar variational principle characterizes the TAM empirical-risk minimizer; in the ridgeless limit it gives a closed-form critical load separating satisfiable and unsatisfiable phases.
Conjecture: Small-tail extrapolation suggests a sharp top-1 threshold at d^2 ~ 2n log n.