The emergence of clusters in self-attention dynamics

Viewing Transformers as interacting particle systems, we describe the geometry of learned representations when the weights are not time dependent. We show that particles, representing tokens, tend to cluster toward particular limiting objects as time tends to infinity. Cluster locations are determined by the initial tokens, confirming context-awareness of representations learned by Transformers. Using techniques from dynamical systems and partial differential equations, we show that the type of limiting object that emerges depends on the spectrum of the value matrix. Additionally, in the one-dimensional case we prove that the self-attention matrix converges to a low-rank Boolean matrix. The combination of these results mathematically confirms the empirical observation made by Vaswani et al. [VSP 17] that leaders appear in a sequence of tokens when processed by Transformers. 1 A classical choice is θ " pW, A, bq P R dˆd ˆRdˆd ˆRd and f θ pxq " W σpAx bq where σ is an elementwise nonlinearity such as the ReLU ([HZRS16b] ).