*Translated and adapted from a Chinese forum post on zhichai.net. Paper: arXiv:2605.05189*
Overview
- Field: Machine Learning
- Authors: Nicholas Barnfield, Juno Kim, Eshaan Nichani, Jason D. Lee, Yue M. Lu
- Published: 2026-05-06
- arXiv: 2605.05189
- The paper asks: how many key-value associations can a d x d linear memory matrix store? The answer depends not only on the d^2 degrees of freedom of the memory matrix, but also on the retrieval criterion.
- In an isotropic Gaussian model for stored pairs, top-1 retrieval — where every signal must beat its largest distractor — requires the logarithmic model-size scale d^2 ~ n log n.
- The correlation matrix memory construction (storing associations by superposing key-target outer products) provably achieves this scale through a sharp phase transition, and the same scaling is necessary for any linear memory. The logarithm is thus the intrinsic extreme-value price of winner-take-all decoding.
- The paper then studies 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, the authors introduce the Tail-Average Margin (TAM), a convex upper-tail criterion that certifies inclusion of the correct target in a controlled candidate list. Under this criterion, capacity follows the quadratic scale d^2 ~ n.
- At load n/d^2 -> alpha, an exact asymptotic theory for the TAM empirical-risk minimizer is developed via a two-parameter scalar variational principle.
- The theory exhibits rich phenomenology:
- In the ridgeless limit, it yields a closed-form critical load separating satisfiable and unsatisfiable phases.
- It predicts the limiting laws of true scores, competitor scores, margins, and percentile profiles.
- A small-tail extrapolation further leads to the conjectural sharp top-1 threshold d^2 ~ 2n log n.
Key Points
Asymptotic Theory
Abstract (Original)
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.
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*Auto-collected on 2026-05-08.*