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

TL;DR

This paper analyzes how attention mechanisms influence signal recovery in sequence models using random matrix theory, revealing phase transitions and optimal weights for improved signal detection.

Contribution

It provides exact spectral characterizations of pooled sequence representations and identifies optimal attention weights for maximizing signal-to-noise ratio.

Findings

Bulk spectrum follows a non-Marchenko--Pastur law due to vocabulary structure.

Signal recovery exhibits two BBP-type phase transitions.

Top eigenvector of R yields optimal attention weights for signal enhancement.

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 $d/V\to\delta$ and $d/N\to\gamma$, 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 $\kappa(MP_\delta\boxtimes MP_\gamma)$, reflecting the finite vocabulary structure. Signal recovery undergoes two successive BBP-type phase transitions characterized by the scalars: $\delta,\gamma,\alpha=w^{\top} R w$ and $\kappa=\|w\|^2$, 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 $\alpha/\kappa$ 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 $\tau/d$ 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.