# Conditional Optimal Bridge for Riemannian Activation Steering

arXiv: https://arxiv.org/abs/2607.10517 · HTML: https://mlsanity.com/paper/2607.10517/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Activation steering methods, one of 13 directions in Mechanistic interpretability.
- Builds on 30 of its 41 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Placed in

- Mechanistic interpretability / [Activation steering methods](https://mlsanity.com/mechinterp/activation-steering-methods/)

## Builds on (most cited first, 30 of 30)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Measuring Massive Multitask Language Understanding](https://mlsanity.com/paper/2009.03300/) (2020)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Representation Engineering: A Top-Down Approach to AI Transparency](https://mlsanity.com/paper/2310.01405/) (2023)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [RealToxicityPrompts: Evaluating Neural Toxic Degeneration in Language Models](https://mlsanity.com/paper/2009.11462/) (2020)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [The Falcon Series of Open Language Models](https://mlsanity.com/paper/2311.16867/) (2023)
- [RewardBench: Evaluating Reward Models for Language Modeling](https://mlsanity.com/paper/2403.13787/) (2024)
- [UltraFeedback: Boosting Language Models with Scaled AI Feedback](https://mlsanity.com/paper/2310.01377/) (2023)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Analyzing the Generalization and Reliability of Steering Vectors](https://mlsanity.com/paper/2407.12404/) (2024)
- [A Spectral Condition for Feature Learning](https://mlsanity.com/paper/2310.17813/) (2023)
- [Controlling Language and Diffusion Models by Transporting Activations](https://mlsanity.com/paper/2410.23054/) (2024)
- [Representation Surgery: Theory and Practice of Affine Steering](https://mlsanity.com/paper/2402.09631/) (2024)
- [Understanding (Un)Reliability of Steering Vectors in Language Models](https://mlsanity.com/paper/2505.22637/) (2025)
- [Steering off Course: Reliability Challenges in Steering Language Models](https://mlsanity.com/paper/2504.04635/) (2025)
- [Controlled LLM Training on Spectral Sphere](https://mlsanity.com/paper/2601.08393/) (2026)
- [Householder Pseudo-Rotation: A Novel Approach to Activation Editing in LLMs with Direction-Magnitude Perspective](https://mlsanity.com/paper/2409.10053/) (2024)
- [The Rogue Scalpel: Activation Steering Compromises LLM Safety](https://mlsanity.com/paper/2509.22067/) (2025)
- [Spherical Steering: Geometry-Aware Activation Rotation for Language Models](https://mlsanity.com/paper/2602.08169/) (2026)
- [ODESteer: A Unified ODE-Based Steering Framework for LLM Alignment](https://mlsanity.com/paper/2602.17560/) (2026)
- [TruthFlow: Truthful LLM Generation via Representation Flow Correction](https://mlsanity.com/paper/2502.04556/) (2025)

## Built on by (most cited first, 2 of 2)

- [Does Steering Break Your Model? A Multi-Dimensional Evaluation Suite for LLM Steering Methods](https://mlsanity.com/paper/2610.07722/) (2026)
- [Steering Geometry: Validating Human Value Geometry in LLM Steering Space](https://mlsanity.com/paper/2609.06289/) (2026)

## Abstract (arXiv)

Activation steering offers a lightweight alternative to fine-tuning for controlling large language models at inference time. While many existing methods implicitly optimize a log-density-ratio objective between desired and undesired activation distributions, they do so heuristically rather than deriving it from a principled optimization problem. Moreover, these methods produce query-independent steering directions that can degrade performance on both in-distribution and out-of-distribution (OOD) inputs. We introduce textscCobras (Conditional Optimal Bridge for Riemannian Activation Steering), which addresses both limitations by casting activation steering as a Schr"odinger Bridge on the residual-stream hypersphere. This formulation yields, to our knowledge, the first principled derivation of the log-density-ratio steering objective from a well-posed optimization problem. Solving the bridge via entropic optimal transport and extracting the probability flow ODE recovers the widely used density-ratio gradient as a special case when the Sinkhorn potentials are uniform. Crucially, the Schr"odinger potentials are evaluated at the current activation, making the resulting steering direction inherently query-adaptive. Empirically, across four models and three alignment axes (helpfulness, truthfulness, and detoxification), textscCobras consistently outperforms prior activation steering baselines while avoiding the OOD degradation commonly observed in existing methods. The code can be found at https://github.com/arshandalili/cobras.