Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior — part 1/6
Cleaned-text code points [0, 15730). Bibliography omitted here; source.md includes it.
# Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior
###### Abstract
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 ℳ h \mathcal{M}_{h} to representations and a behavior manifold ℳ y \mathcal{M}_{y} to output probability distributions. We then test the link ℳ h ↔ ℳ y \mathcal{M}_{h}\leftrightarrow\mathcal{M}_{y} via interventions: we find that steering along ℳ h \mathcal{M}_{h} , which we term manifold steering , yields behavioral trajectories that follow ℳ y \mathcal{M}_{y} , 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 ℳ y \mathcal{M}_{y} recovers activation trajectories that trace the curvature of ℳ h \mathcal{M}_{h} . 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 .
## 1 Introduction
A plethora of geometric structures have been documented in neural network representations ( Modell et al., 2025b ; Park et al., 2025b ; Kozlowski et al., 2025 ; Shai et al., 2024b ; Pearce et al., 2025 ; Gurnee et al., 2026 ) . Recent literature has begun to identify the origins of these structures by attributing them back to data statistics shaped by conceptual structure ( Karkada et al., 2026 ; Prieto et al., 2026 ; Park et al., 2025b ; Merullo et al., 2025 ) . However, we have barely begun to understand what causal role these geometric structures play in a model’s computation (cf. Engels et al. 2024 ; Kantamneni and Tegmark 2025 ; Csordás et al. 2024 ; Sarfati et al. 2026 ). We address this question by intervening on model activations under different geometric assumptions and measuring the effect on behavior.
Currently, it is common for activation-based intervention methods to assume a Euclidean geometry for activation space, where steering is performed by adding a steering vector to model activations with a scalar that modulates intervention strength ( Bau et al., 2019 ; Subramani et al., 2022 ; Marks and Tegmark, 2024 ; Panickssery et al., 2024 ; Turner et al., 2024 ; Li et al., 2023 ; Rimsky et al., 2024 ; Chen et al., 2025 ) . This approach is motivated by the linear representation hypothesis (LRH), which posits that neural activations can be decomposed into atomic concepts encoded along single (approximately) orthogonal directions ( Smolensky, 1986 ; Park et al., 2023 ; Elhage et al., 2022b ) . However, linear steering often produces degraded fluency, diversity collapse, and unstable off-target behavior ( Wu et al., 2025 ; Da Silva et al., 2025 ; Bigelow et al., 2025 ; Tan et al., 2024 ; Hao et al., 2025 ; Bhalla et al., 2024 ; Pres et al., 2024 ) , which suggests the assumed Euclidean geometry is inappropriate.
In this work, we advance the hypothesis that representation geometry provides a blueprint for effective steering that will overcome the limitations of the linear approach. Steering is fundamentally about how internal representations control behavior, so to test this hypothesis we must study not only paths through activation space, but also the behavioral trajectories induced by interventions along these paths. Successful steering will produce trajectories that are in line with the model’s natural (unintervened) output distribution. If we are right, then interventions that respect the geometry of internal representations and interventions that respect the geometry of behavior will be one and the same. Motivated by this, we make the following contributions in this work.
• Uncovering isometric geometries in neural network representation and behavior. We use tasks where models output a distribution over a set of concepts with known structure. In each task, we fit an activation manifold ℳ h \mathcal{M}_{h} to internal representations and a behavior manifold ℳ y \mathcal{M}_{y} to model outputs (probability distributions over task-relevant concepts). We show the two geometries are tightly interlinked via a scaled isometry relation: geodesic distances on ℳ h \mathcal{M}_{h} align closely with those on ℳ y \mathcal{M}_{y} , and neither match Euclidean distances.
• Validating the causal role of representation geometry. We perform geometry-aware steering experiments and compare against the baseline of linear steering (see Fig. 1 ). We show linear steering cuts through low-density regions of behavior space and passes through unnatural intermediate distributions; meanwhile, steering along the activation manifold ℳ h \mathcal{M}_{h} yields behavioral trajectories that follow ℳ y \mathcal{M}_{y} closely. In fact, optimizing for paths along ℳ y \mathcal{M}_{y} recovers activation trajectories that trace the curvature of ℳ h \mathcal{M}_{h} , further tightening the link between activation geometry and behavior.
• A theoretical framework for geometry-aware steering. Building on the results above, we formulate steering as a problem of choosing the right geometry for activation space, rather than the right direction . In particular, we argue steering can be defined as the problem of finding a geodesic connecting two points under different activation-space metrics: linear steering assumes a flat metric (Euclidean geometry), steering along the activation manifold uses a metric derived from natural activations, and steering optimized to follow the behavior manifold uses a metric derived from natural behaviors.
We demonstrate these findings hold across modalities and tasks. In large language models, we test geometries from cyclic concepts (weekdays, months; Engels et al. 2024 ; Modell et al. 2025b ), sequential concepts (ages, letters), and multi-dimensional graph structures learned in context ( Park et al., 2025b ) . In a video world model, we test a geometry of physical position in a simulated environment (mountain car; Moore 1990 ; Towers et al. 2024 ). Together, these findings provide evidence for the posited account, and support steering along neural manifolds as the principled form of activation-based intervention.
## 2 The Geometry of Representation and Behavior
### 2.1 Setup
#### Running example.
We will explicate our framework and empirical methods using a running example where a language model is required to reason about the days of the week ( Engels et al., 2024 ) . Specifically, we consider prompts of the form: What day is k k days after z z ? with z ∈ 𝒵 = { Mon , Tue , … , Sun } z\in\mathcal{Z}=\{\mathrm{Mon},\mathrm{Tue},\ldots,\mathrm{Sun}\} and k ∈ { 1 , … , 7 } k\in\{1,\dots,7\} . Given such a prompt, the LM outputs a probability distribution over all possible tokens.
#### Concept geometry.
We draw inspiration from work on conceptual spaces in cognitive science, where conceptual domains, e.g., days of the week, are geometrically enriched with a metric such that distances between points encode similarity and guide patterns of inference ( Shepard 1987 ; Gärdenfors 2000 ; Tenenbaum and Griffiths 2001 ; Bellmund et al. 2018 ; see Fel et al. 2025b ; Lubana et al. 2025 ; Yocum et al. 2025 ; Modell et al. 2025b for related work in interpretability). For example, the days of the week 𝒵 \mathcal{Z} may be organized in a cyclic structure that is captured by a metric d 𝒵 d_{\mathcal{Z}} measuring temporal distance between days, e.g., neighboring days are closer together. Indeed, when humans mistakenly report the current day, they most often confuse it with its neighboring days ( Ellis et al., 2015 ) .
Karkada et al. (2026) and Prieto et al. (2026) show that similarity structure between days of the week is reflected in the statistics of training data, which in turn shape the geometry of internal representations ( Engels et al., 2024 ; Park et al., 2025a ; Modell et al., 2025a ; Prieto et al., 2026 ) . We hypothesize that a model’s output distributions over 𝒵 \mathcal{Z} are similarly shaped by d 𝒵 d_{\mathcal{Z}} : e.g., when asked What day is four days after Monday? , the model concentrates mass on Friday and spreads the remainder onto nearby days like Thursday and Saturday.
#### Notation.
We work with two spaces: the activation space 𝒜 = ℝ n \mathcal{A}=\mathbb{R}^{n} , and the behavior space 𝒴 = Δ | 𝒵 | \mathcal{Y}=\Delta^{|\mathcal{Z}|} , which is the open probability simplex 1 1 1 We require the open simplex { 𝒑 ∈ ℝ > 0 | 𝒵 | : ∑ i p i = 1 } \{\bm{p}\in\mathbb{R}^{|\mathcal{Z}|}_{>0}:\sum_{i}p_{i}=1\} , i.e., strictly positive entries. The closed simplex, which includes faces where some p i = 0 p_{i}=0 , has boundary and corners and is not a smooth manifold. over the conceptual domain 𝒵 \mathcal{Z} , with an additional ‘other’ class for off-concept probability mass. For an input x x , let 𝒑 ( x ) ∈ 𝒴 \bm{p}(x)\in\mathcal{Y} denote the model’s output distribution over 𝒵 \mathcal{Z} , given by restricting the full vocabulary distribution of the model to the tokens in 𝒵 \mathcal{Z} , in addition to the ‘other’ class for remaining probability mass. Let 𝒉 ( x ) ∈ 𝒜 \bm{h}(x)\in\mathcal{A} denote an activation vector of interest for input x x .
For a class of input queries that share the same answer, e.g., What is two days after Monday? and What is three days after Sunday? , we average the hidden activations and output distributions to produce “activation centroids” and “behavior centroids”, respectively.
#### Experimental tasks.
We perform language model experiments on four tasks, two with cyclic conceptual structure and two with sequential conceptual structure. The cyclic tasks require reasoning about days of the week and months of the year, e.g., What is four months after January? . The sequential tasks require reasoning about letters and ages, e.g., What is four letters after m? or Alice is 7, Bob is 5 years older. How old is Bob? .
### 2.2 Fitting the Manifolds
We fit a smooth manifold within each space to the model’s unintervened activations or outputs for a task: ℳ h ⊆ 𝒜 \mathcal{M}_{h}\subseteq\mathcal{A} , the activation manifold , and ℳ y ⊆ 𝒴 \mathcal{M}_{y}\subseteq\mathcal{Y} , the behavior manifold . To fit the activation manifold ℳ h \mathcal{M}_{h} , we reduce activation vectors 𝒉 ( x ) \bm{h}(x) to 64 dimensions via PCA, compute “concept centroids” (e.g., averaging all activations where the correct answer is Wednesday), and fit cubic splines ( Reinsch, 1967 ) through the centroids (see App. A.3 for further spline fitting details). To fit the behavior manifold ℳ y \mathcal{M}_{y} , we follow a similar procedure but first map each centroid from the probability simplex onto Hellinger space via p ↦ p p\mapsto\sqrt{p} . This linearizes the geometry of the simplex: the Hellinger distance between distributions becomes an ordinary Euclidean distance, d H ( p , q ) = 1 2 ‖ p − q ‖ d_{H}(p,q)=\tfrac{1}{\sqrt{2}}\|\sqrt{p}-\sqrt{q}\| , so we can fit splines and compare distributions with standard Euclidean tools while still respecting the underlying probabilistic geometry ( Amari and Nagaoka, 2000 ) . Decoded points are squared back to recover valid distributions (further details, including how we keep the fit on the sphere, are in App. A.4 ). Unless stated otherwise, we use Llama 3.1 8B ( Touvron et al., 2023 ) with activations from layer 28, and visualize manifolds via 3 3 D PCA.
### 2.3 Conceptual Structure Appears in Behavior and Activation Space
We will now begin our investigation into the connection between the activation and behavior manifolds. Before we perform any interventions on internal representations, we examine the structural correspondence between these spaces, and measure whether distances along the two manifolds are proportional (a scaled isometry).
We find that both the activation and behavior manifolds recapitulate conceptual structure (see Figs. 2 , 3 for visualizations). For example, in the case of days of the week, the activations and output distributions are arranged in order around a loop, with Monday adjacent to Tuesday and Sunday, and Thursday on the opposite side (Fig. 2 ). The circle representing days of the week in activation space is already known to exist ( Engels et al., 2024 ; Modell et al., 2025b ; Karkada et al., 2026 ) . However, the circle in behavior space is a novel discovery, and results from sharply-peaked output distributions placing most mass on the target concept, with the remainder concentrated on its neighbors. The correspondence is striking; output distributions and internal activations recover the same cyclic ordering. In contrast, the conceptual structure for the ages and letters task is sequential rather than cyclic, and so both the activations and output distributions for these tasks lie on an open curve (Fig. 3 ).
Going beyond qualitative structural correspondence, we wish to examine the mapping between distances along each manifold. We test this by computing pairwise distances between points in both spaces: geodesic distances d ℳ h ( m i , m j ) d_{\mathcal{M}_{h}}(m_{i},m_{j}) on the activation manifold, and geodesic distances d ℳ y ( p i , p j ) d_{\mathcal{M}_{y}}(p_{i},p_{j}) on the behavior manifold. We compute geodesic distance on ℳ h \mathcal{M}_{h} using cumulative Euclidean distance between points along a geodesic path, and follow the same procedure for geodesic distance on ℳ y \mathcal{M}_{y} , but using cumulative Hellinger distance (see App. A.5 for further details). The two distances are highly correlated ( r = 0.99 r=0.99 weekdays, r = 0.89 r=0.89 months, r = .999 r=.999 letters, r = .999 r=.999 ages) indicating that ℳ h \mathcal{M}_{h} and ℳ y \mathcal{M}_{y} are approximately isometric. Meanwhile, linear paths between the same activation-space points correlate less well with ℳ y \mathcal{M}_{y} geodesics, with the relationship showing clear non-linear patterns ( r = 0.89 r=0.89 weekdays, r = 0.53 r=0.53 months, r = 0.71 r=0.71 letters, r = 0.36 r=0.36 ages).
This correspondence leads to an intuitive hypothesis. ℳ y \mathcal{M}_{y} was fit to unintervened task behavior, so it traces a path through natural output distributions for the model. If the ℳ h ↔ ℳ y \mathcal{M}_{h}\leftrightarrow\mathcal{M}_{y} mapping holds, paths along one manifold should track paths along the other. Interventions in activation space that follow ℳ h \mathcal{M}_{h} should produce natural trajectories along ℳ y \mathcal{M}_{y} . Conversely, activation-space paths that are optimized to produce trajectories on ℳ y \mathcal{M}_{y} should recover ℳ h \mathcal{M}_{h} . Next, we test both directions via intervention: representation to behavior ( ℳ h → ℳ y \mathcal{M}_{h}\rightarrow\mathcal{M}_{y} ) and behavior to representation ( ℳ h ← ℳ y \mathcal{M}_{h}\leftarrow\mathcal{M}_{y} ).
## 3 Connecting Representation and Behavior via Intervention