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

1 Introduction

• Common activation-based interventions add a steering vector under an assumed Euclidean geometry, and linear steering can degrade fluency, diversity, and off-target stability.Source: §1 · pp. 1–2
• Activation and behavior manifolds are tightly linked by a scaled isometry: their geodesic distances align closely, unlike Euclidean distances.The authors fit manifolds to internal representations and output probability distributions over task-relevant concepts.Source: §1 · p. 2
• Steering along activation manifold Mh yields behavioral trajectories following My, while linear steering crosses low-density regions and produces unnatural intermediate distributions.The paper contrasts paths derived from the intrinsic geometry of fitted manifolds with straight-line Euclidean paths.Source: §1 · p. 2, Figure 1 · p. 2
• Optimizing activation-space interventions to follow My recovers trajectories that trace the curvature of Mh.Source: §1 · p. 2
• The authors frame steering as finding geodesics under different activation-space metrics: Euclidean, derived from natural activations, or derived from natural behaviors.Source: §1 · pp. 2–3
• The authors test the account across language-model tasks with cyclic, sequential, and graph geometries, and a video world model task involving physical dynamics.Source: §1 · p. 3

2.1 Setup

• The framework represents inputs with activation vectors in A = Rn and output distributions in Y, the open probability simplex over concept tokens plus an ‘other’ class.Queries sharing an answer can be summarized by averaging their activations and output distributions into activation and behavior centroids.Source: §2.1 · p. 3

2.2 Fitting the Manifolds

• For four language-model tasks, the authors fit cubic-spline manifolds to PCA-reduced activation centroids and to output centroids mapped into Hellinger space.The tasks cover cyclic reasoning about weekdays and months and sequential reasoning about letters and ages; the default model is Llama 3.1 8B with layer-28 activations.Source: §2.1 · p. 3, §2.2 · p. 4

2.3 Conceptual Structure Appears in Behavior and Activation Space

• Both spaces recover the tasks’ conceptual structure: weekday and month concepts form cycles, while age and letter concepts lie on open sequential curves.For weekdays, behavior distributions place most mass on the target and the remainder on neighboring days, yielding a behavior-space circle that the authors describe as a novel discovery.Source: Figure 2 · p. 4, §2.3 · p. 5, Figure 3 · p. 5
• Geodesic distances correlate strongly across activation and behavior manifolds (r = 0.99 weekdays, r = 0.89 months, r = .999 letters, r = .999 ages), unlike linear activation paths (r = 0.89 weekdays, r = 0.53 months, r = 0.71 letters, r = 0.36 ages).The activation-space geodesics use cumulative Euclidean distance; behavior-space geodesics use cumulative Hellinger distance.Source: Figure 2 · p. 4, Figure 3 · p. 5, §2.3 · p. 6
• The approximate activation–behavior manifold correspondence motivates bidirectional intervention tests: manifold-following activation paths should yield natural behavior paths, and behavior-targeted paths should recover activation-manifold trajectories.The authors test representation-to-behavior and behavior-to-representation directions.Source: §2.3 · p. 6

3.1 Steering Intervention Notation

• An intervention replaces an activation at a chosen layer with a target, and a steering path interpolates between endpoint activations using either activation-space or manifold coordinates.Manifold steering maps interpolated intrinsic coordinates back to activations, keeping the path on the fitted activation manifold; linear steering interpolates directly in activation space.Source: §3.1 · p. 6

3.2 Steering Along the Activation Manifold Follows the Behavior Manifold

• 2.8× lower cumulative output energy on average shows manifold steering produces more natural paths than linear steering across four tasks.Manifold steering versus linear steering yielded weekdays EBC = 0.34 ± 0.03 vs. 0.93 ± 0.11; months EBC = 0.36 ± 0.01 vs. 1.09 ± 0.06; letters 2.42 ± 0.07 vs. 6.95 ± 0.27; and ages EBC = 5.21 ± 0.09 vs. 13.49 ± 0.29, with all statistical comparisons p < 0.001. Figure 4 shows smooth, ordered transitions along adjacent concepts, unlike linear steering’s jumps between non-adjacent concepts.Source: Figure 4 · p. 7, §3.2 · p. 8

3.3 Behavior Space Geometry Recovers the Activation Manifold

• Pullback paths optimized to follow the behavior manifold recover activation-manifold paths more closely than linear steering, supporting a bidirectional geometry correspondence.For weekdays, R2_pullback = 0.75 ± 0.04 versus R2_linear = 0.32 ± 0.05; all reported statistical comparisons yield p < 0.001.Source: Figure 5 · p. 8, §3.3 · p. 9

3.4 Unifying Steering Strategies Through Geometry

• The framework distinguishes Euclidean linear steering, density-derived manifold steering, and output-derived pullback steering as paths under different activation-space geometries.Density geometry makes off-manifold regions expensive and on-manifold movement cheap; pullback geometry uses the behavior-manifold metric through the activation-to-behavior Jacobian, with regularization preserving positive definiteness. The paper proposes that these geometries converge on approximately the same activation paths, suggesting an approximate Riemannian isometry.Source: §3.4 · pp. 9–10

In-context learning tasks with synthetic conceptual spaces

• The experiments extend prior one-dimensional conceptual spaces to in-context tasks where arbitrary tokens map to grid or cylinder graphs and random walks provide input sequences.The graph defines conceptual distance, and the fitted manifold has two intrinsic coordinates that control independent dimensions of the space.Source: §4 · p. 10

Manifold fitting

• The fitted ICLR activation manifold is a two-dimensional surface whose semi-spherical shape reflects more frequent random-walk visits to inner sites; thin plate splines fit activation and behavior surfaces.The manifold is topologically a surface with no holes or tears; centroids are computed from last-token activations by graph location.Source: §4 · pp. 10–11

Manifold vs. Linear steering

• Manifold steering yields smooth transitions along one grid dimension while preserving the other, enabling factored control; linear steering instead teleports between endpoints.The result extends smooth, ordered manifold-steering transitions to multidimensional spaces learned in context.Source: Figure 6 · p. 11, §4 · p. 12

Isometry results

• Geodesic distances on activation and behavior manifolds correlate at r = .99 for the 5 × 5 grid and 9 × 9 cylinder; linear-path correlations are 0.90 and 0.81, respectively.MDS embeddings show grid structure for both manifolds, while linear activation distances yield a warped surface.Source: Figure 6 · p. 11, §4 · p. 11

• The Mountain Car model predicts next frames from prior frames and actions using a convolutional encoder, 16-dimensional action embeddings, a 64-unit GRU, and a residual-image decoder [Moore, 1990; Sutton & Barto, 2018].Source: §5 · p. 12
• From 100 rollouts, encoder activations form a curved, low-dimensional position manifold, while position-indexed output distributions define a second one-dimensional behavior manifold.Both manifolds are parameterized by position; the behavior manifold uses Hellinger coordinates, and the curves close because the wall and goal map to neighboring activations.Source: §5 · p. 12
• Manifold steering yields smooth, coherent car movement, whereas linear steering produces blurred intermediate placements and teleportation; optimizing output-space paths recovers activation paths tracing Mh.Linear steering also spreads the predicted position distribution more than on-manifold paths.Source: Figure 7 · p. 13, §5 · p. 13
• On-manifold activation and behavior distances recover position ordering, while linear distances scramble it; their arc-length correlation is r = 0.99 versus r = 0.06 for linear activation paths and behavior arc-length.The activation manifold folds back in ambient encoder space, bringing activations for distant positions arbitrarily close.Source: §5 · pp. 13–14, Figure 7 · p. 13

Activation Steering and the Linear Representation Hypothesis

• Linear steering is motivated by LRH’s direction-based concept geometry, but outside that setting its higher-order output moments can be unconstrained and adversarial (Sarfati et al., 2026).Prior work found linear steering can match the first moments of current and target output distributions; the effects on higher-order moments may help explain incoherent outputs.Source: §6 · p. 14
• Geometry-respecting steering smoothly interpolates source and target distributions, extending beyond narrow examples that lack a general account of representation-to-behavior mapping or a toy softmax model.The narrow-domain examples directly write a days-of-the-week circle or follow a helix representing numbers; Park et al. (2026) study a toy model with a simple softmax mapping.Source: §6 · p. 14

Activation Geometry and its Origins

• Prior work finds concepts encoded in curved low-dimensional geometries across modalities and architectures, with newer accounts linking data statistics and symmetries to representation geometry at scale.Toy-setting work connected data-generating structure to representation geometry; symmetry-based accounts propose geometries that reflect distributional uncertainty and may explain in-context representations.Source: §6 · p. 14

Causal Analysis of Neural Networks

• Critiques argue that probing and visualization alone can mislead about behavior, motivating activation interventions grounded in causal mediation and causal abstraction frameworks.These frameworks connect interpretability research to theories of causality.Source: §6 · pp. 14–15

Discussion

• The authors establish a causal bridge: activation-manifold distances align with behavior-manifold distances, and geodesic steering in either space traces geodesics in the other.Steering along activation-manifold geodesics induces smooth, coherent behavioral transitions; optimizing interventions for behavioral geodesics recovers activation trajectories that follow the activation manifold.Source: §7 · p. 15
• Linear steering pathologies are attributed to a mismatch between assumed flat geometry and the curved geometry of representation space, not an inherent limitation of intervention.The discussion names brittleness, incoherence, and off-target effects as pathologies and reframes steering from finding the right direction to finding the right geometry. It cites Wu et al. (2025), Bigelow et al. (2025), Da Silva et al. (2025), Bhalla et al. (2024), and Tan et al. (2024) in this context.Source: §7 · p. 15
• Although the origins of shared geometry are unstudied, synthetic in-context-learning tasks suggest conceptual structure can constrain representation and behavior beyond training-data statistics.These tasks assign tokens contextual meanings unlike those learned during training, yet the model forms novel representations and behaviors with shared geometry; the discussion interprets this as behavior computationally constrained by activation geometry and notes Yocum et al. (2025) formalize the claim.Source: §7 · p. 15
• The paper proposes intrinsic coordinates on fitted activation manifolds as causal-analysis units, with naturally occurring activations constraining against degenerate solutions.This proposal addresses the search for primitive units of representation and the concern that arbitrarily complex causal units permit degenerate solutions.Source: §7 · p. 15

Expanding experimental validation to more complex domains

• Future validation should test abstract concepts such as refusals (Arditi et al., 2024), sycophancy (Vennemeyer et al., 2025), and persuasion (Costello et al., 2026), whose output geometry may be subtler.The right representational primitives for these concepts are uncertain, and complex scenarios will require isolating conceptual geometry from in-the-wild data rather than synthetic, template-based text.Source: §8 · p. 16

Moving from token to sequence-level outputs

• Extending from next-token distributions to sequences may require a belief-based account, with geometry-aware steering changing latent output properties rather than concrete token-level behavior.The proposed formalization treats beliefs as latent variables underlying a model’s posterior predictive (Bigelow et al., 2023, 2025; Wurgaft et al., 2025).Source: §8 · p. 16

Fitting the geometry

• Manifold fitting must yield an operator for navigating the manifold; unsupervised fitting could broaden applicability beyond domains with ground-truth relations that define intrinsic coordinates.The paper used a specific fitting protocol, while its analyzed domains supplied ground-truth relations for spline fitting.Source: §8 · p. 16

Manipulating intermediate algorithmic variables

• Future control protocols should manipulate intermediate algorithmic variables that mediate information flow from input to output, not only output behavior directly.One example is an image model determining an object’s shape in service of predicting its weight.Source: §8 · p. 16

A Experimental Details for Language Tasks

• Appendix §A describes the procedures for the experiments in Sections §2–§4; Appendix §B covers the mountain-car experiment in §5.Source: A Experimental Details for Language… · p. 26

A.1 Tasks and Datasets

• Datasets pair cyclic weekdays/months and filtered sequential letters/ages addition with Park et al. (2025b)’s ICLR graph-walk tasks on 5×5 grids and 9×9 cylinders.ICLR prompts are random walks of 2048 entity tokens sampled without backtracking.Source: §A.1 · p. 26, Table 1 · p. 26

A.2 Model, Intervention Site, and Output Distribution

• Experiments use Llama 3.1 8B (Touvron et al., 2023) layer-28 last-token residual-stream interventions; outputs aggregate spelling-variant probability mass and an “other” bin.The resulting output distribution has |Z| + 1 classes.Source: §A.2 · p. 26

A.3 Fitting the Activation Manifold Mh

• Activations are projected into a 64-dimensional PCA subspace, where centroid-interpolating cubic splines use ordinal coordinates for sequential tasks and unsupervised angular coordinates for cyclic tasks.Sequential tasks use natural cubic splines, while cyclic tasks use periodic cubic splines.Source: §A.3 · pp. 26–27
• For ICLR, a thin-plate spline interpolates centroids using graph coordinates, with ghost points enforcing closure along the cylinder’s periodic dimension.The cylinder combines a linear and a periodic dimension.Source: §A.3 · p. 27

A.4 Fitting the Behavior Manifold My

• Behavior centroids are square-root embedded in Hellinger space, then spline-fit in a sphere’s tangent plane and decoded by the exp-map, keeping My on-sphere and passing through centroids.The tangent-plane fit uses the log-map to project centroids and the exponential map to decode spline outputs.Source: §A.4 · p. 27

A.5 Geodesic Distances and the Isometry Test

• Geodesic distances are approximated by decoding 150 intrinsic-coordinate subdivisions, and isometry is Pearson correlation of pairwise distances after adding between-centroid vertices while excluding same-geodesic pairs.The added interior vertices are decoded onto the manifolds to probe geometry between centroids.Source: §A.5 · pp. 27–28

A.6 Steering Interventions

• Manifold steering decodes 50 geodesic waypoints on Mh while preserving off-subspace residuals; linear steering replaces the full residual stream along straight centroid segments.The study uses 16 base prompts for natural-domain tasks and 5 for ICLR tasks, with up to 50 randomly sampled concept pairs per task.Source: §A.6 · p. 28

A.7 Naturalness Metric

• Naturalness energy sums Bhattacharyya distances from induced outputs at 50 steering waypoints to their closest points on the behavior manifold.The distance stays within the manifold’s Hellinger geometry because D_BC = −log(1 − d_H^2).Source: §A.7 · p. 28
• For each centroid pair, the metric averages cumulative waypoint distances across base prompts, then reports the mean and standard error across pairs.There are up to 50 sampled centroid pairs.Source: §A.7 · p. 28

A.8 Pullback Optimization

• Pullback optimization targets a behavior-manifold spline geodesic between behavior centroids, sampled at 20 uniform fractions, and fits an activation path to match it.The induced behavioral trajectory is compared with the target distributions at those same waypoints.Source: §A.8 · p. 28
• The activation path is a natural cubic spline with 10 control vectors optimized in the first 32 components of a 64-dimensional PCA subspace.The remaining 32 components and orthogonal residual stay at base-prompt activation values; comparison paths span the full 64-dimensional subspace.Source: §A.8 · p. 28
• The squared Hellinger losses are averaged over 16 freshly sampled base prompts per pair conditioned on ground-truth za, then minimized with L-BFGS and strong-Wolfe line search at 10 control-t positions.It runs 50 outer steps with up to 5 inner iterations each and stops when consecutive-step relative loss change falls below 10^-3.Source: §A.8 · p. 29

A.9 Pullback Recovery R2

• Pullback recovery measures each pullback waypoint’s orthogonal closest-point distance to the activation path after projection into its variance SVD basis.The passage also specifies projecting both paths into that basis.Source: §A.9 · p. 29
• The recovery comparison uses the straight chord between activation centroids in the 64-dimensional PCA subspace, with mean ± standard error and paired t-tests against that baseline.This baseline is not the post-intervention linear-steering trajectory.Source: §A.9 · p. 29

B Experimental Details for the Vision Task.

• Section B provides additional experimental details for the vision-task experiment described in §5.Source: B Experimental Details for the Vision… · p. 30

B.1 Mountain Car.

• MountainCar activations were collected from 100 rollouts under mixed stochastic policies, then labeled with ground-truth position to fit the encoder’s position manifold.Episodes used a noisy momentum policy with probability 0.7 or an oscillating square-wave policy with probability 0.3.Source: §B.1 · p. 30
• For MountainCar, a spline through 100 position bins tracked the activation data manifold while the endpoint chord cut through its interior; Ridge probes recovered position at R2 ≈ 0.95 and velocity at R2 ≈ 0.90.The behavior manifold was fitted as a position-parameterized spline through Hellinger-coordinate output centroids.Source: §B.1 · pp. 30–31

C.1 In-Context Learning of Representations.

• With 2048 tokens of context, Llama 3.1 8B exceeds 80% neighborhood accuracy on a 9 × 9 cylinder, and factored control generalizes beyond the graph domain.Neighborhood accuracy is the probability mass on valid neighbors.Source: Figure 9 · p. 32, §C.1 · p. 32
• On the grid and cylinder tasks, manifold steering paths closely follow the behavior manifold, while linear steering shows teleportation and warps structural distances.Manifold distances in activation space show a clear structural match to behavior-space distances.Source: Figure 9 · p. 32, Figure 10 · p. 33

C.2 Manifold steering allows manipulation of uncertainty without loss of structure.

• Manifold steering along weekday-transition circles maintains their order while entropy increases across addition-value groups, suggesting a controlled handle on model confidence.The groups form concentric circles along a second, cylinder-like manifold dimension; the experiment used Llama 3.1 70B, layer 70.Source: §C.2 · p. 34, Figure 11 · p. 34

C.3 Mountain Car

• Arc-length distances along Mh and My correlated at r = 0.996 across 1,225 pairs, while activation chord and behavior arc length correlated at r = 0.06.The chord cuts across the encoder loop rather than following its conceptual geometry.Source: §C.3 · p. 35
• For each of 30 endpoint pairs, the study optimized 30 free activation waypoints, initialized on the linear chord, to match a conformal behavior target on the simplex.The waypoints were optimized jointly via L-BFGS with strong-Wolfe line search.Source: §C.3 · pp. 35–36
• Across 30 endpoint pairs, mean Euclidean distance to Mh was 0.29 for pullback, versus 2.22 for linear chord and 0.20 for geometric (Mh); pullback recovered 95.4% and beat chords on 30/30 pairs.Degradation was concentrated on pairs with an endpoint near the extreme wall (p ≈ −1.2); on the remaining ∼20 pairs, pullback was essentially indistinguishable from Mh.Source: §C.3 · p. 36
• At α = 0, pullback follows the unrestricted Hellinger geodesic and leaves Mh; as α grows, the behavior target moves onto My and the recovered path tracks Mh.Figure 12 shows the pullback-optimized activation path converging from its chord initialization onto Mh, while its behavior-space pushforward tracks My.Source: Figure 12 · p. 35, §C.3 · p. 36