Spectral Selection in Symmetric Self-Attention Dynamics
MCML Authors
Christian Kühn
Prof. Dr.
Collaborating PI
Abstract
Christian Kühn
Prof. Dr.
Collaborating PI
Abstract
We study self-attention dynamics on the unit sphere as an interacting particle system arising from an idealized Transformer-type update. Under a symmetry assumption on weight matrices given by $Q^⊤ K=V=V ^⊤$, the flow admits a gradient-flow structure and an exact reformulation in the eigenbasis of V, revealing a spectral mode-selection mechanism. We show that the dynamics exhibits two distinct asymptotic scenarios: homogeneous alignment toward the dominant eigendirection when one positive eigenvalue strictly dominates all others in modulus, and sign-split polarization toward the most negative eigendirection when V is negative definite. In particular, we obtain local stability criteria for pure-mode equilibria and global selection results in both regimes. These results provide a rigorous finite-particle description of how the spectrum of the weight matrices organizes asymptotic patterns in a symmetric self-attention flow, and highlight how the symmetric setting renders the dynamics amenable to mathematical analysis.
BibTeXKey: KY26