The Routing and Filtering Structure of Attention

2026-05-12

AI Summary

This work addresses the entanglement of routing and filtering functions in conventional attention mechanisms, which leads to structural opacity and optimization challenges. The authors propose S-D Attention, which explicitly decouples the interaction matrix into a low-rank routing component and a symmetric filtering component, thereby clearly distinguishing these two mechanisms for the first time. They further uncover that routing self-organizes into a spectral cascade phenomenon in deep networks. Leveraging this insight, they achieve stable training without layer normalization and validate their approach using linear attention variants (e.g., ELU+1) and effective rank analysis. Experiments show that linearizing the first seven layers of a 125M-parameter model incurs less than 5% perplexity degradation, while cascade-informed architectures reduce attention parameters by 47%–65% with only a 3.9%–8.4% increase in perplexity.

Abstract

The attention interaction matrix $QK^{\top}$ contains two entangled computations: a skew-symmetric component that redistributes information between positions (routing) and a symmetric component that scales mutual relevance (filtering). We decompose 1776 heads across five pretrained transformers and find routing operating at low rank, well below the routing capacity allocated by the weight kernel. We introduce $S$-$D$ attention as a diagnostic parameterization that disentangles routing from filtering by construction with guaranteed stability ($\mathrm{Re}(λ) \le 0$) and trains stably without layer normalization. When disentangled and unnormalized, routing self-organizes into a spectral cascade, effective rank $2$ at the first layer, expanding with depth across six scales from 7M to 355M parameters. The cascade predicts where attention can be simplified: linearizing the first seven layers of 125M $S$-$D$ attention costs ${<}5\%$ perplexity, whereas standard attention collapses under the same intervention. The linearizable region widens with depth. Replacing the first four layers with ELU+1 linear attention reaches within $1.4\%$ of baseline at full head dimension. Cascade-allocated architectures trade attention parameters for perplexity ($47\%-65\%$ fewer attention parameters at $+3.9\%$ to $+8.4\%$ PPL). The routing-filtering decomposition makes the spectral budget legible; the cascade makes it actionable.

Problem

• attention
• routing
• filtering
• spectral cascade
• attention decomposition

Innovation

• routing-filtering decomposition
• S-D attention
• spectral cascade
• linearizable attention
• low-rank routing