Manifold Steering Reveals the Shared Geometry of Neural Network Representation and Behavior — part 2/6
Cleaned-text code points [15730, 31707). Bibliography omitted here; source.md includes it.
Now that we have established a correlational correspondence between the activation and behavior manifolds across four tasks, we turn to steering interventions for causal evidence. First, we steer along ℳ h \mathcal{M}_{h} and measure whether output trajectories follow ℳ y \mathcal{M}_{y} (§ 3.2 ). Second, we optimize interventions on internal representations to produce output distributions that follow ℳ y \mathcal{M}_{y} and measure whether the optimized activation trajectory follows ℳ h \mathcal{M}_{h} (§ 3.3 ).
### 3.1 Steering Intervention Notation
The basic intervention operation entails replacing the model’s activation at a chosen layer with a target activation, and continuing the forward pass. Given a base input x x and a target 𝒉 ⋆ ∈ 𝒜 \bm{h}^{\star}\in\mathcal{A} , we write 𝒑 𝒉 ← 𝒉 ⋆ ( x ) \bm{p}_{\bm{h}\leftarrow\bm{h}^{\star}}(x) for the resulting output distribution. A steering path is a curve 𝝅 : [ 0 , 1 ] → 𝒜 \bm{\pi}:[0,1]\to\mathcal{A} between endpoints 𝒉 0 ⋆ \bm{h}^{\star}_{0} and 𝒉 1 ⋆ \bm{h}^{\star}_{1} , inducing a trajectory 𝒑 𝒉 ← 𝝅 ( t ) ( x ) \bm{p}_{\bm{h}\leftarrow\bm{\pi}(t)}(x) through behavior space 𝒴 \mathcal{Y} . The behavioral trajectory will be non-stationary only if the target 𝐡 \mathbf{h} mediates the causal effect from input to output ( Pearl, 2001 ; Vig et al., 2020 ; Mueller et al., 2024 ) .
We consider two strategies, both constructed by interpolation between the endpoints; the strategies differ only in the coordinate system in which the interpolation is taken (Fig. 1 ): 𝝅 lin ( t ) \displaystyle\bm{\pi}_{\mathrm{lin}}(t) = ( 1 − t ) 𝒉 0 ⋆ + t 𝒉 1 ⋆ \displaystyle=(1{-}t)\,\bm{h}^{\star}_{0}+t\,\bm{h}^{\star}_{1} (linear steering); (1) 𝝅 m ( t ) \displaystyle\bm{\pi}_{\mathrm{m}}(t) = 𝒔 ( ( 1 − t ) 𝒖 0 + t 𝒖 1 ) , 𝒖 i = 𝒔 − 1 ( 𝒉 i ⋆ ) \displaystyle=\bm{s}\bigl((1{-}t)\,\bm{u}_{0}+t\,\bm{u}_{1}\bigr),\quad\bm{u}_{i}=\bm{s}^{-1}(\bm{h}^{\star}_{i}) (manifold steering). (2) In the above, 𝒔 : ℝ k → 𝒜 \bm{s}:\mathbb{R}^{k}\to\mathcal{A} is a parameterization of ℳ h \mathcal{M}_{h} —the map sending k k -dimensional intrinsic coordinates to the corresponding point on the manifold in the activation space 𝒜 \mathcal{A} . Linear steering (also known as ‘diff-in-means steering’) ( Bau et al., 2018 ; Subramani et al., 2022 ; Turner et al., 2023 ) interpolates in 𝒜 \mathcal{A} directly—the standard additive-vector baseline. Manifold steering interpolates in the intrinsic coordinates of ℳ h \mathcal{M}_{h} and maps the result back through 𝒔 \bm{s} , so 𝝅 m \bm{\pi}_{\mathrm{m}} stays on the activation manifold ℳ h \mathcal{M}_{h} throughout. Each strategy thus corresponds to a different choice of geometry on activation space, which we concretize in § 3.4 .
### 3.2 Steering Along the Activation Manifold Follows the Behavior Manifold
For every pair of start and end values, e.g., from Tuesday to Friday, we steer from the start centroid to the end centroid in activation space using manifold and linear steering with K = 50 K=50 intervention points along each path. We report the average trajectory in behavior space over a set of 16 prompts sampled randomly from the task’s input distribution (Fig. 4 , see App. A.6 for further experimental details). We find that manifold steering produces smooth and ordered behavioral transitions: probability mass shifts steadily through adjacent values of a concept—from Monday to Tuesday to Wednesday to Thursday—while linear steering instead exhibits ‘teleportation’ : mass jumps between non-adjacent concepts as the straight line cuts through the manifold’s interior.
This qualitative evidence is encouraging, but we have yet to examine our key hypothesis that interventions along the activation manifold ℳ h \mathcal{M}_{h} produce natural output trajectories that follow ℳ y \mathcal{M}_{y} . This would mean that outputs produced under manifold steering resemble those produced without intervention. We quantify this via an “energy function”, as described next, under which a natural trajectory is one of low cumulative energy as defined by ℳ y \mathcal{M}_{y} .
#### An Energy-based View of Naturalness.
Energy functions have a long history in machine learning as a way to measure plausibility under a model ( Hopfield, 1982 ; LeCun et al., 2006 ) . These functions assign low values for likely states and high values for unlikely ones, with the standard correspondence E ( 𝒙 ) ∝ − log p ( 𝒙 ) E(\bm{x})\propto-\log p(\bm{x}) giving energy the interpretation of an unnormalized log-density ( Hopfield, 1982 ; LeCun et al., 2006 ; Grathwohl et al., 2019 ; Song and Kingma, 2021 ; Béthune et al., 2025 ) . We adopt the same view here. The model’s output distributions on unintervened forward passes trace out a low-energy region of behavior space (approximately captured by the manifold ℳ y \mathcal{M}_{y} ) and a steering trajectory is natural to the extent it stays within that region. Concretely, given a steering path 𝝅 : [ 0 , 1 ] → 𝒜 \bm{\pi}:[0,1]\to\mathcal{A} , let 𝜸 ( t ) = 𝒑 𝒉 ← 𝝅 ( t ) ( 𝒙 ) \bm{\gamma}(t)=\bm{p}_{\bm{h}\leftarrow\bm{\pi}(t)}(\bm{x}) be the behavioral trajectory it induces. We define its cumulative output energy: E BC ( 𝜸 ) = ∫ 0 1 d BC ( 𝜸 ( t ) , ℳ y ) 𝑑 t , E_{\text{BC}}(\bm{\gamma})\;=\;\int_{0}^{1}d_{\text{BC}}\!\bigl(\bm{\gamma}(t),\,\mathcal{M}_{y}\bigr)\,dt, (3) where d BC ( 𝒑 , ℳ y ) = inf 𝒒 ∈ ℳ y d BC ( 𝒑 , 𝒒 ) = − log ( ∑ i 𝒑 i 𝒒 i ) d_{\text{BC}}(\bm{p},\mathcal{M}_{y})=\inf_{\sqrt{\bm{q}}\in\mathcal{M}_{y}}d_{\text{BC}}(\bm{p},\bm{q})=-\log(\sum_{i}\sqrt{\bm{p}_{i}}\sqrt{\bm{q}_{i}}) is the Bhattacharyya distance to the nearest point on ℳ y \mathcal{M}_{y} , a natural choice given that it is simply the negative log of the dot product in Hellinger space (in which ℳ y \mathcal{M}_{y} is fit). We note that the formulation above is a tractable proxy we use to estimate distance from the model’s natural output distribution, yet it is but one instantiation of a more general framework we develop in § 3.4 . Applying this measure, we find that manifold steering (weekdays E BC = 0.34 ± 0.03 E_{\text{BC}}=0.34\pm 0.03 ; months E BC = 0.36 ± 0.01 E_{\text{BC}}=0.36\pm 0.01 ; letters 2.42 ± 0.07 2.42\pm 0.07 ; ages E BC = 5.21 ± 0.09 E_{\text{BC}}=5.21\pm 0.09 ) produces significantly more natural paths, i.e., lower cumulative energy, than linear steering (weekdays E BC = 0.93 ± 0.11 E_{\text{BC}}=0.93\pm 0.11 ; months E BC = 1.09 ± 0.06 E_{\text{BC}}=1.09\pm 0.06 ; letters 6.95 ± 0.27 6.95\pm 0.27 ; ages E BC = 13.49 ± 0.29 E_{\text{BC}}=13.49\pm 0.29 ); on average, we see an improvement of a factor of 2.8 × 2.8\times , with all statistical comparisons yielding p < 0.001 p<0.001 .
We find further verification of the claim above by visualizing output trajectories in behavior space (Fig. 4 ). Manifold steering consistently traces paths close to ℳ y \mathcal{M}_{y} , while linear steering cuts through regions far from the behavior manifold, yielding less natural outputs. This result provides causal support for the correspondence between activation and output geometry, and establishes manifold steering as a principled form of steering that yields natural behavioral trajectories ( ℳ h → ℳ y \mathcal{M}_{h}\rightarrow\mathcal{M}_{y} ). Next, we explore whether we can find evidence from the opposite direction ( ℳ y → ℳ h \mathcal{M}_{y}\rightarrow\mathcal{M}_{h} ).
### 3.3 Behavior Space Geometry Recovers the Activation Manifold
In this section, we aim to uncover whether steering intervention paths optimized to follow the behavior manifold ℳ y \mathcal{M}_{y} recover the activation manifold ℳ h \mathcal{M}_{h} . To do so, we first take a path π y ∗ \pi_{y}^{*} along ℳ y \mathcal{M}_{y} in behavior space. We work within the layer and the first 32 dimensions of the subspace in which we fit the activation manifold ( 64 64 dimensional PCA), and optimize via L-BFGS for a path in activation space which, upon intervention, induces the behavioral path π y ∗ \pi_{y}^{*} (see App. A.8 for further details regarding the optimization procedure). We call the resulting path in activation space the pullback .
To quantify how faithfully a pullback path in activation space π h pullback \pi_{h}^{\text{pullback}} recapitulates the manifold steering path π h ∗ \pi_{h}^{*} , we report an intrinsic R 2 R^{2} . Both paths are projected into a common subspace given by the singular directions explaining 99 % 99\% of the variance in π h ∗ \pi_{h}^{*} , restricting the comparison to directions where the path actually extends. We then compute the R 2 R^{2} in this subspace, defining the residual at each point of π h pullback \pi_{h}^{\text{pullback}} as its orthogonal closest-point distance to π h ∗ \pi_{h}^{*} . We compute this score for each optimized pullback path π h pullback \pi_{h}^{\text{pullback}} and compare with a linear path baseline (see App. A.9 for more details).
Results are shown in Fig. 5 . We find that the pullback activation paths follow the activation manifold ℳ h \mathcal{M}_{h} more closely than the linear steering path, and resemble the shape of the manifold steering paths of § 3.2 (weekdays R pullback 2 = 0.77 ± 0.03 R^{2}_{\text{pullback}}=0.77\pm 0.03 vs. R linear 2 = 0.42 ± 0.07 R^{2}_{\text{linear}}=0.42\pm 0.07 ; months R pullback 2 = 0.75 ± 0.04 R^{2}_{\text{pullback}}=0.75\pm 0.04 vs. R linear 2 = 0.32 ± 0.05 R^{2}_{\text{linear}}=0.32\pm 0.05 ; ages R pullback 2 = 0.47 ± 0.05 R^{2}_{\text{pullback}}=0.47\pm 0.05 vs. R linear 2 = 0.24 ± 0.01 R^{2}_{\text{linear}}=0.24\pm 0.01 ; letters R pullback 2 = 0.78 ± 0.04 R^{2}_{\text{pullback}}=0.78\pm 0.04 vs. R linear 2 = 0.23 ± 0.03 R^{2}_{\text{linear}}=0.23\pm 0.03 . All statistical comparisons yield p < 0.001 p<0.001 ). Again we see a striking correspondence between representation and behavior; despite being derived from different sources—manifold steering from the density of activations and pullback from the structure of outputs—the two geometries are tightly connected.
Taken together, these results, alongside those of § 3.2 , provide bidirectional support for the connection between activation geometry and behavior. This convergence indicates that ℳ h \mathcal{M}_{h} is a core object in the model’s representation: the geometry of activation space and the geometry of behavior are alternate views of the same underlying conceptual organization.
### 3.4 Unifying Steering Strategies Through Geometry
In the sections above, we analyzed three methods for steering between two points in activation space that each assume a different geometry: linear steering, which assumes a flat Euclidean geometry; manifold steering, which derives a geometry from naturally occurring activations; and pullback steering, which derives a geometry from naturally occurring output distributions. We provided empirical support for our hypothesis that the geometries derived from internal activations and output behaviors are much more similar to each other than the standard Euclidean geometry. We now formalize the question of how to steer as how to choose the right geometry for activation space .
#### The Geometry of Steering:
Consider a Riemannian metric 𝑮 \bm{G} , which assigns an inner product at each point of 𝒜 \mathcal{A} ; together with a path 𝝅 : [ 0 , 1 ] → 𝒜 \bm{\pi}:[0,1]\to\mathcal{A} , this defines the notion of path length as follows. L 𝑮 ( 𝝅 ) = ∫ 0 1 𝝅 ˙ ( t ) ⊤ 𝑮 ( 𝝅 ( t ) ) 𝝅 ˙ ( t ) 𝑑 t . L_{\bm{G}}(\bm{\pi})\;=\;\int_{0}^{1}\sqrt{\dot{\bm{\pi}}(t)^{\top}\,\bm{G}(\bm{\pi}(t))\,\dot{\bm{\pi}}(t)}\,dt. (4) Then, a geodesic is defined as the path of minimum length between two endpoints, and each choice of geometry picks out a steering strategy. The strategies of linear steering and manifold steering (§ 3.2 ), written as interpolations in two different coordinate systems (Eqs. 1 , 2 ), are two such choices; the pullback procedure of § 3.3 is a third. Now, we make all three geometries explicit.
###### Definition 1 (Geometries of Steering) .
Let E : 𝒜 → ℝ E:\mathcal{A}\to\mathbb{R} be an energy function such that E ( 𝐡 ) ∝ − log p ( 𝐡 ) E(\bm{h})\propto-\log p(\bm{h}) , and let 𝐠 y \bm{g}_{y} be a chosen Riemannian metric on ℳ y \mathcal{M}_{y} . We define: 𝑮 I \displaystyle\bm{G}_{I} = 𝑰 n , \displaystyle=\bm{I}_{n}, (linear steering) (5) 𝑮 E ( 𝒉 ) \displaystyle\bm{G}_{E}(\bm{h}) = ( α e − E ( 𝒉 ) + β ) − 1 𝑰 n , \displaystyle=\bigl(\alpha\,e^{-E(\bm{h})}+\beta\bigr)^{-1}\bm{I}_{n}, (manifold steering) (6) 𝑮 F ( 𝒉 ) \displaystyle\bm{G}_{F}(\bm{h}) = 𝑱 𝑭 ( 𝒉 ) ⊤ 𝒈 y ( 𝑭 ( 𝒉 ) ) 𝑱 𝑭 ( 𝒉 ) + ϵ 𝑰 n , \displaystyle=\bm{J}_{\bm{F}}(\bm{h})^{\top}\,\bm{g}_{y}\bigl(\bm{F}(\bm{h})\bigr)\,\bm{J}_{\bm{F}}(\bm{h})+\epsilon\,\bm{I}_{n}, (pullback) (7) where α , β > 0 \alpha,\beta>0 are calibration constants, ϵ > 0 \epsilon>0 regularizes the pullback, 𝐅 : 𝒜 → 𝒴 \mathbf{F}:\mathcal{A}\to\mathcal{Y} is the function from naturally occurring activations to naturally occurring behaviors, and 𝐠 y \bm{g}_{y} is any Riemannian metric on ℳ y \mathcal{M}_{y} (e.g., the induced Hellinger metric used in our experiments).
We discuss the intuitive interpretation of Defn. 1 below.
• The Flat Geometry G I \bm{G}_{I} . Linear steering treats activation space as Euclidean: all directions and regions are equally valid, with Geodesics as straight lines ℓ ( t ) = ( 1 − t ) 𝒉 0 + t 𝒉 1 \bm{\ell}(t)=(1{-}t)\,\bm{h}_{0}+t\,\bm{h}_{1} . This geometry thus encodes no knowledge of naturally occurring activation or outputs.
• The Density Geometry G E \bm{G}_{E} . Manifold steering derives a geometry for activation space from naturally occurring internal representations. Specifically, consider the geometry induced from an energy function E ( 𝒉 ) ∝ − log p ( 𝒉 ) E(\bm{h})\propto-\log p(\bm{h}) by rescaling the identity according to local density. Here e − E ( 𝒉 ) e^{-E(\bm{h})} plays the role of an unnormalized density: large where activations concentrate (on ℳ h \mathcal{M}_{h} ) and small where they are sparse (off ℳ h \mathcal{M}_{h} ). The inverse makes off-manifold regions expensive and on-manifold movement cheap, with constants α , β > 0 \alpha,\beta>0 calibrating the dynamic range ( Béthune et al., 2025 ) . Geodesics under 𝑮 E \bm{G}_{E} thus follow ℳ h \mathcal{M}_{h} , recovering manifold steering.
• The Pullback Geometry G F \bm{G}_{F} . The steering path given by pullback derives geometric structure from naturally occurring model outputs. Specifically, 𝑮 F \bm{G}_{F} is the pullback of a chosen geometry on ℳ y \mathcal{M}_{y} through the Jacobian of the map from activation space to behavior space 𝑭 : 𝒜 → 𝒴 \bm{F}:\mathcal{A}\to\mathcal{Y} . By construction, path length under 𝑮 F \bm{G}_{F} equals path length of the induced behavioral trajectory along ℳ y \mathcal{M}_{y} (up to a regularization term). Geodesics under 𝑮 F \bm{G}_{F} are therefore activation paths whose induced behavioral trajectories are geodesics on ℳ y \mathcal{M}_{y} —exactly the pullback construction of § 3.3 . The regularization ϵ 𝑰 n \epsilon\,\bm{I}_{n} ensures positive definiteness, since 𝑱 𝑭 \bm{J}_{\bm{F}} has rank at most | 𝒵 | − 1 ≪ n |\mathcal{Z}|-1\ll n ; as ϵ \epsilon tends to 0, the geometry approaches the pure pullback in the range of 𝑱 𝑭 \bm{J}_{\bm{F}} and remains Euclidean in its null space.
Overall, we claim that while the metrics 𝑮 E \bm{G}_{E} and 𝑮 F \bm{G}_{F} are derived from different sources (internal activations and outputs, respectively), they converge on approximately the same paths in activation space (§ 3.3 ). This suggests the manifolds ℳ h \mathcal{M}_{h} and ℳ y \mathcal{M}_{y} are two images of the same conceptual geometry, related by an approximate Riemannian isometry. Consequently, the question of optimally steering model behavior boils down to isolating the geometry of a concept and defining operators to navigate it.
## 4 Manifold Steering Yields Factored Control in Multi-Dimensional Spaces