# GEOSTEER: Geodesic Optimization for Activation Steering in Large Language Models

arXiv: https://arxiv.org/abs/2609.10658 · HTML: https://mlsanity.com/paper/2609.10658/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Activation steering methods, one of 13 directions in Mechanistic interpretability.
- Builds on 21 of its 29 references that are in the map.
- First posted on arXiv 9 Sep 2026.

## Placed in

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

## Builds on (most cited first, 21 of 21)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [Root Mean Square Layer Normalization](https://mlsanity.com/paper/1910.07467/) (2019)
- [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 Falcon Series of Open Language Models](https://mlsanity.com/paper/2311.16867/) (2023)
- [In-context Vectors: Making In Context Learning More Effective and Controllable Through Latent Space Steering](https://mlsanity.com/paper/2311.06668/) (2023)
- [Programming Refusal with Conditional Activation Steering](https://mlsanity.com/paper/2409.05907/) (2024)
- [ReFT: Representation Finetuning for Language Models](https://mlsanity.com/paper/2404.03592/) (2024)
- [Improving Instruction-Following in Language Models through Activation Steering](https://mlsanity.com/paper/2410.12877/) (2024)
- [Controlling Language and Diffusion Models by Transporting Activations](https://mlsanity.com/paper/2410.23054/) (2024)
- [Semantics-Adaptive Activation Intervention for LLMs via Dynamic Steering Vectors](https://mlsanity.com/paper/2410.12299/) (2024)
- [Angular Steering: Behavior Control via Rotation in Activation Space](https://mlsanity.com/paper/2510.26243/) (2025)
- [Representation Surgery: Theory and Practice of Affine Steering](https://mlsanity.com/paper/2402.09631/) (2024)
- [Multi-Attribute Steering of Language Models via Targeted Intervention](https://mlsanity.com/paper/2502.12446/) (2025)
- [Householder Pseudo-Rotation: A Novel Approach to Activation Editing in LLMs with Direction-Magnitude Perspective](https://mlsanity.com/paper/2409.10053/) (2024)
- [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)
- [Differentially Private Steering for Large Language Model Alignment](https://mlsanity.com/paper/2501.18532/) (2025)
- [Steering Information Utility in Key-Value Memory for Language Model Post-Training](https://mlsanity.com/paper/2507.05158/) (2025)

## Abstract (arXiv)

Activation steering provides a lightweight way to control large language models (LLMs) by modifying their hidden activations at inference time. Among these approaches, norm-preserving steering aims to change model behavior without altering the activation norm, reducing the risk of representation collapse and degradation. However, existing norm-preserving methods are limited by predefined steering trajectories and by their reliance on one-step updates, which may fail to capture the complex structure of activation distributions. We propose GeoSteer, an optimization-based method for norm-preserving activation steering. GeoSteer formulates steering as a Riemannian optimization problem and updates activations through a sequence of small geodesic steps on the representation manifold. To avoid fixed steering directions, GeoSteer learns a nonlinear activation-space objective that distinguishes desired from undesired activations, and uses this function to adaptively guide each steering step. This multistep formulation yields smoother, more stable, and more consistent steering behavior while preserving the activation norm. Across TruthfulQA, RealToxicityPrompts, and UltraFeedback benchmarks, GeoSteer consistently improves over state-of-the-art activation steering baselines. These results suggest that norm-preserving steering can be made more effective by replacing predefined one-step edits with adaptive, geometry-aware optimization.