# Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior

arXiv: https://arxiv.org/abs/2605.05115 · HTML: https://mlsanity.com/paper/2605.05115/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 88 of its 125 references that are in the map.
- First posted on arXiv 6 May 2026.

## Builds on (most cited first, 30 of 88)

- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Locating and Editing Factual Associations in GPT](https://mlsanity.com/paper/2202.05262/) (2022)
- [π0: A Vision-Language-Action Flow Model for General Robot Control](https://mlsanity.com/paper/2410.24164/) (2024)
- [Dream to Control: Learning Behaviors by Latent Imagination](https://mlsanity.com/paper/1912.01603/) (2019)
- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Inference-Time Intervention: Eliciting Truthful Answers from a Language Model](https://mlsanity.com/paper/2306.03341/) (2023)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (2022)
- [Steering Llama 2 via Contrastive Activation Addition](https://mlsanity.com/paper/2312.06681/) (2023)
- [Refusal in Language Models Is Mediated by a Single Direction](https://mlsanity.com/paper/2406.11717/) (2024)
- [Mass-Editing Memory in a Transformer](https://mlsanity.com/paper/2210.07229/) (2022)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [Steering Language Models With Activation Engineering](https://mlsanity.com/paper/2308.10248/) (2023)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [The Linear Representation Hypothesis and the Geometry of Large Language Models](https://mlsanity.com/paper/2311.03658/) (2023)
- [The Geometry of Truth: Emergent Linear Structure in Large Language Model Representations of True/False Datasets](https://mlsanity.com/paper/2310.06824/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Dissecting Recall of Factual Associations in Auto-Regressive Language Models](https://mlsanity.com/paper/2304.14767/) (2023)
- [Causal Abstractions of Neural Networks](https://mlsanity.com/paper/2106.02997/) (2021)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [Persona Vectors: Monitoring and Controlling Character Traits in Language Models](https://mlsanity.com/paper/2507.21509/) (2025)
- [Function Vectors in Large Language Models](https://mlsanity.com/paper/2310.15213/) (2023)
- [Direct and Indirect Effects](https://mlsanity.com/paper/1301.2300/) (2013)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [LEACE: Perfect linear concept erasure in closed form](https://mlsanity.com/paper/2306.03819/) (2023)
- [Extracting Latent Steering Vectors from Pretrained Language Models](https://mlsanity.com/paper/2205.05124/) (2022)
- [Understanding Neural Networks through Representation Erasure](https://mlsanity.com/paper/1612.08220/) (2016)
- [AxBench: Steering LLMs? Even Simple Baselines Outperform Sparse Autoencoders](https://mlsanity.com/paper/2501.17148/) (2025)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)

## Built on by (most cited first, 26 of 26)

- [FishBack: Pullback Fisher Geometry for Optimal Activation Steering in Transformers](https://mlsanity.com/paper/2605.17231/) (2026)
- [Structuring Sparsity: Block-Sparse Featurizers Capture Visual Concept Manifolds](https://mlsanity.com/paper/2606.25234/) (2026)
- [Stories in Space: In-Context Learning Trajectories in Conceptual Belief Space](https://mlsanity.com/paper/2605.12412/) (2026)
- [When Does Activation Steering Change What a Model Computes From?](https://mlsanity.com/paper/2606.29522/) (2026)
- [SMIXAE: Towards Unsupervised Manifold Discovery in Language Models](https://mlsanity.com/paper/2605.09224/) (2026)
- [Riemannian-Manifold Steering: Geometry-Aware Generative Autoencoders for Label-Free Steering](https://mlsanity.com/paper/2605.24942/) (2026)
- [The ACUTE Protocol: Operationalizing Language Model Activations for Better Calibration, Utility, and Trust](https://mlsanity.com/paper/2606.07822/) (2026)
- [Capsule Lens: Locating and Tracking Concept Geometry in Model Representations](https://mlsanity.com/paper/2609.05575/) (2026)
- [Relational Rank Geometry in Transformers: Detecting and Steering Hidden-State Relation Frames](https://mlsanity.com/paper/2605.29634/) (2026)
- [LaRA: Layer-wise Representation Analysis for Detecting Data Contamination in RL Post-Training](https://mlsanity.com/paper/2605.29888/) (2026)
- [Harnessing the Latent Space: From Steering Vectors to Model Calibrators for Control and Trust](https://mlsanity.com/paper/2607.00083/) (2026)
- [A Shared Subcircuit Lets LLMs Count Down Across Tasks](https://mlsanity.com/paper/2607.12279/) (2026)
- [Context Is King: How In-Context Specification Shapes the Geometry of Concepts](https://mlsanity.com/paper/2607.24425/) (2026)
- [What, Where, and How: Probing Spatiotemporal Representations in Video Foundation Models](https://mlsanity.com/paper/2609.01551/) (2026)
- [MAxBench: A Multinomial Concept Recovery Benchmark](https://mlsanity.com/paper/2609.13072/) (2026)
- [A Chosen Future Can Still Be Rewritten: Causal Writability in Video Models](https://mlsanity.com/paper/2609.15980/) (2026)
- [World Modeling in Transformers](https://mlsanity.com/paper/2609.21748/) (2026)
- [Activation Flow: Manufacturing Activations for Steering](https://mlsanity.com/paper/2609.32530/) (2026)
- [BOReFT: Manifold Steering of Language Models for Black-box Optimization](https://mlsanity.com/paper/2609.33722/) (2026)
- [Causal and Interpretable Structures in LLM Compositional Tasks](https://mlsanity.com/paper/2609.35970/) (2026)
- [RolloutFaith: Auditing Persistent Internal Interventions in Visual World Model](https://mlsanity.com/paper/2609.36843/) (2026)
- [Temporally-Resolved Token Attribution Reveals the Generation Dynamics of Diffusion Language Models](https://mlsanity.com/paper/2610.01177/) (2026)
- [Beyond Linear Concepts: Discovering and Aligning Non-Linear Concept Manifolds in Large Language Models](https://mlsanity.com/paper/2610.01821/) (2026)
- [Language Model Activations Inhabit Privileged Error-Correcting Basins](https://mlsanity.com/paper/2610.04183/) (2026)
- [Extracting Persona Subspaces Through Iterative Nullspace Projection For Modulation](https://mlsanity.com/paper/2610.04676/) (2026)
- [Beyond the Linear Representation Hypothesis: Non-Linear Activation Steering in Text-to-Image Models](https://mlsanity.com/paper/2610.06945/) (2026)

## Abstract (arXiv)

Neural representations carry rich geometric structure; but does that structure causally shape behavior? To address this question, we intervene along paths through activation space defined by different geometries, and measure the behavioral trajectories they induce. In particular, we test whether interventions that respect the geometry of activation space will yield behaviors close to those the model exhibits naturally. Concretely, we first fit an activation manifold Mh to representations and a behavior manifold My to output probability distributions. We then test the link Mh leftrightarrow My via interventions: we find that steering along Mh, which we term manifold steering, yields behavioral trajectories that follow My, while linear steering -- which assumes a Euclidean geometry -- cuts through off-manifold regions and hence produces unnatural outputs. Moreover, optimizing interventions in activation space to produce paths along My recovers activation trajectories that trace the curvature of Mh. We demonstrate this bidirectional relationship between the geometry of representation and behavior across tasks and modalities. In language models, we use reasoning tasks with cyclic and sequential geometries as well as in-context learning tasks with more complex graph geometries. In a video world model, we use a task with geometry corresponding to physical dynamics. Overall, our work shows that geometry in neural representation is not merely incidental, but is in fact the proper object for enabling principled control via intervention on internals. This recasts the core problem of steering from finding the right direction to finding the right geometry.