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

arxiv:2605.05115

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Representation alignment · 2026-05-06

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Read in parts

1. Part 1
2. Part 2
3. Part 3
4. Part 4
5. Part 5
6. Part 6

Contents

• Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior
• Abstract
• 1 Introduction
• 2 The Geometry of Representation and Behavior
• 2.1 Setup
• Running example.
• Concept geometry.
• Notation.
• Experimental tasks.
• 2.2 Fitting the Manifolds
• 2.3 Conceptual Structure Appears in Behavior and Activation Space
• 3 Connecting Representation and Behavior via Intervention
• 3.1 Steering Intervention Notation
• 3.2 Steering Along the Activation Manifold Follows the Behavior Manifold
• An Energy-based View of Naturalness.
• 3.3 Behavior Space Geometry Recovers the Activation Manifold
• 3.4 Unifying Steering Strategies Through Geometry
• The Geometry of Steering:
• Definition 1 (Geometries of Steering) .
• 4 Manifold Steering Yields Factored Control in Multi-Dimensional Spaces
• In-context learning tasks with synthetic conceptual spaces.
• Manifold fitting.
• Isometry results.
• Manifold vs. Linear steering.
• 5 Manifold Steering on a Visual World Model: Mountain Car Task
• Environment and model architecture.
• Activation and behavior manifold fitting.
• Geometry-aware steering in activation space.
• The geometry assumed by linear steering is not faithful to the conceptual ordering.
• 6 Related Work
• Activation Steering and the Linear Representation Hypothesis.
• Activation Geometry and its Origins.
• Causal Analysis of Neural Networks.
• 7 Discussion
• Geometry-aware steering reveals the shared structure of behavior and representation.
• Where does the shared geometry of behavior and representation come from?
• Intrinsic coordinates of representation manifolds as units of causal analysis.
• 8 Future Work and Limitations
• Acknowledgments
• Appendix A Experimental Details for Language Tasks
• A.1 Tasks and Datasets
• Natural domain tasks.
• In-context learning of representations.
• A.2 Model, Intervention Site, and Output Distribution
• A.3 Fitting the Activation Manifold ℳ h \mathcal{M}_{h}
• A.4 Fitting the Behavior Manifold ℳ y \mathcal{M}_{y}
• A.5 Geodesic Distances and the Isometry Test
• A.6 Steering Interventions
• A.7 Naturalness Metric
• A.8 Pullback Optimization
• Path Parameterization.
• Loss and optimizer.
• A.9 Pullback Recovery R 2 R^{2}
• Appendix B Experimental Details for the Vision Task
• B.1 Mountain Car.
• Data collection.
• Manifold fitting.
• Appendix C Additional Results
• C.1 In-Context Learning of Representations.
• C.2 Manifold steering allows manipulation of uncertainty without loss of structure.
• C.3 Mountain Car
• Structural Correspondence between ℳ h \mathcal{M}_{h} and ℳ y \mathcal{M}_{y} .
• Pullback: Behavior space steering.
• Instructions for reporting errors