The Laplacian Mechanism Improves Transformers by Reshaping Token Geometry

Transformers leverage attention, the residual connection, and layer normalization to control the variance of token representations. We propose to modify attention into a Laplacian mechanism that gives the model more direct control over token variance. We conjecture that this helps transformers achieve the ideal token geometry. To investigate our conjecture, we first show that incorporating the Laplacian mechanism into transformers induces consistent improvements across benchmarks in computer vision and language. Next, we study how the Laplacian mechanism impacts the geometry of token representations using various tools: 1) principal component analysis, 2) cosine similarity metric, 3) analysis of variance, and 4) Neural Collapse metrics. Our investigation shows that the Laplacian mechanism reshapes token embeddings toward a geometry of maximal separability: tokens collapse according to their classes, and the class means exhibit Neural Collapse.