Geometric Analysis of Token Selection in Multi-Head Attention

Abstract: We present a geometric framework for analysing multi-head attention in LLMs. Without altering the mechanism, we view standard attention through a top-N selection lens and study its behaviour directly in value-state space. We define geometric metrics - Precision, Recall, and F-score - to quantify separability between selected and non-selected tokens, and derive non-asymptotic bounds with explicit dependence on dimension and margin under empirically motivated assumptions (stable value norms with a compressed sink token, exponential similarity decay, and piecewise attention weight profiles). The theory predicts a small-N operating regime of strongest non-trivial separability and clarifies how sequence length and sink similarity shape the metrics. Empirically, across LLaMA-2-7B, Gemma-7B, and Mistral-7B, measurements closely track the theoretical envelopes: top-N selection sharpens separability, sink similarity correlates with Recall. We also found that in LLaMA-2-7B heads specialize into three regimes - Retriever, Mixer, Reset - with distinct geometric signatures. Overall, attention behaves as a structured geometric classifier with measurable criteria for token selection, offering head level interpretability and informing geometry-aware sparsification and design of attention in LLMs.