How Does Attention Help? Insights from Random Matrices on Signal Recovery from Sequence Models

• URL: http://arxiv.org/abs/2605.06826v1
• Date: Thu, 07 May 2026 18:28:01 GMT
• Title: How Does Attention Help? Insights from Random Matrices on Signal Recovery from Sequence Models
• Abstract summary: We derive exact characterizations of the limiting eigenvalue distribution, outlier eigenvalues, and eigenvector alignment with the hidden signal.<n>We show that parameter-free causal self-attention with $/d$ score scaling yields deterministic harmonic weights that improve signal recovery over mean pooling.
• Score: 6.758792133077052
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the spectral properties of sample covariance matrices constructed from pooled sequence representations, where token embeddings are drawn from a fixed two-class Gaussian mixture table and pooled via (fixed) attention weights. Working in the high-dimensional regime $d,V,N\to\infty$ with $d/V\toδ$ and $d/N\toγ$, we derive exact characterizations of the limiting eigenvalue distribution, outlier eigenvalues, and eigenvector alignment with the hidden signal. The bulk spectrum follows a non-Marchenko--Pastur law given by the free multiplicative convolution $κ(MP_δ\boxtimes MP_γ)$, reflecting the finite vocabulary structure. Signal recovery undergoes two successive BBP-type phase transitions characterized by the scalars: $δ,γ,α=w^{\top} R w$ and $κ=\|w\|^2$, where $w$ denotes the attention pooling weights and $R$ the positional correlation matrix. An aftermath of our analysis demonstrates that the optimal attention weights maximizing the signal-to-noise ratio $α/κ$ are given by the (normalized) top eigenvector of $R$, and we show (as a particular case of our analysis) that parameter-free causal self-attention with $τ/d$ score scaling yields deterministic harmonic weights that improve signal recovery over mean pooling whenever early tokens carry more signal. Extensive simulations confirm sharp agreement between theory and finite-dimensional experiments.

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