Stream separation improves Bregman conditioning in transformers

TL;DR

This paper investigates how stream separation techniques enhance the Bregman geometry conditioning in transformer layers, improving the reliability of linear interventions by addressing the curvature in representation space.

Contribution

It introduces a controlled experimental framework to measure intermediate layer geometry and demonstrates that stream separation significantly improves Bregman metric conditioning in transformers.

Findings

Stream separation increases effective rank by up to 22.

Per-layer supervision has a smaller impact on conditioning.

Cosine similarity predicts steering effectiveness with a threshold near 0.3.

Abstract

Linear methods for steering transformer representations, including probing, activation engineering, and concept erasure, implicitly assume the geometry of representation space is Euclidean. Park et al. [Park et al., 2026] showed that softmax induces a curved Bregman geometry whose metric tensor is the Hessian of the log-normalizer, $H({\lambda}) = Cov[{\gamma} | {\lambda}]$. Ignoring this curvature causes Euclidean steering to leak probability mass to unintended tokens. Their analysis applies at the output layer. We measure this Hessian at intermediate layers in a controlled 2x2 design crossing stream separation with per-layer supervision (vocabulary decoding loss at each layer), all at matched vocabulary and parameter count. In standard single-stream transformers, H is severely degenerate at intermediate layers (effective rank 8 in 516 dimensions). Stream separation improves conditioning by up to 22 in effective rank, even without auxiliary supervision. Per-layer supervision helps, but less. The cosine similarity between primal and dual concept directions predicts per-layer steering effectiveness on downstream tasks, with a threshold near 0.3. These results bear on the reliability of linear safety interventions, which depend on the geometry being well-conditioned at the layer where they are applied.