Spectral Degeneration of Self-Attention under Isotropic Score Geometry
This paper introduces an idealized spectral model for the pre-softmax score matrix in transformers, motivated by the rotational isotropy of query and key representations. The authors prove that the row-wise softmax map transforms this structure into a degenerate attention matrix where dominant modes are preserved while bulk spectral distributions concentrate at zero. The stable rank remains bounded for bidirectional attention but grows logarithmically for causal attention, explaining how attention compresses non-dominant directions.
Why it matters: Provides a rigorous theoretical explanation for how self-attention preserves dominant directions while compressing bulk modes, offering fundamental insights into transformer behavior.