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The Linear Representation Hypothesis for Vision-Language-Action Models

Minseok Jeong, Hyewon Choi, Hiroyasu Tsukamoto, SooJean Han

arXiv:2609.30996v2cs.LGcs.AI

TL;DR

VLA representations, actions, and physical dynamics are coupled, while the extension of the LRH to this setting remains an open question. The paper develops a signature-based framework for action-conditioned probing and steering, with an oracle construction and pretrained-VLA experiments supporting its predicted mechanisms.

  • Problem

    Whether the LRH extends to VLA models remains largely unexplored, despite the potential to use representations when fine-tuning is expensive and demonstrations are costly.

    Source: §1 · pp. 1, 3

  • Method

    The paper develops a control-theoretic LRH formulation with signature-based action-conditioned probes and a signature generalized linear model for stochastic action chunks.

    Source: §1 · pp. 2–3, §4 · p. 5, §5 · p. 6

  • Results

    An oracle construction verifies the predicted mechanisms, while OpenVLA-OFT experiments find linear probing and steering competitive with more flexible nonlinear alternatives.

    Source: §1 · p. 2, §7 · p. 10

  • Takeaways & Limitations

    The results support a framework in which propagated physical quantities can be probed and expected outcomes steered through linear structure in representations and policies.

    Source: §7 · p. 10

    Identifying a representation naturally suited to both probing and steering remains open, so the experiments are primarily a proof of concept.

    Source: §7 · p. 10

Abstract

from arXiv · show

The linear representation hypothesis (LRH) has become a standard lens for measuring and intervening on semantic information through the internal representations of large language models (LLMs). A growing body of work has begun extending this perspective to vision-language-action (VLA) models, but the dynamical nature of embodied interaction introduces an additional challenge. Unlike semantic attributes commonly studied in LLMs, such as gender or language, a physical quantity of interest (QoI) in a VLA evolves jointly with the system dynamics: the representation influences the actions selected by the policy, which alter the physical state and, in turn, the next representation. In this paper, we develop a theoretical, signature-based formulation of the LRH for VLA that unifies representations and policies. On the representation side, we establish the existence of representations from which the future evolution of a QoI under a candidate action trajectory can be recovered via linear probing. On the policy side, we introduce a signature generalized linear model for stochastic action chunks. This structure yields a monotonic change in the expected future QoI along linear paths in natural parameter space, enabling linear steering. We first validate this structure in a controlled oracle setting, then examine whether the same probing and steering mechanisms emerge in a pretrained VLA.

1 INTRODUCTION

The LRH offers a linear interface for probing and steering model representations, but VLA control couples representations to actions and evolving physical quantities. The paper develops a control-theoretic, signature-based formulation and tests its predicted mechanisms in an oracle construction and a pretrained VLA.

1 INTRODUCTION

  • VLA representations offer a lower-cost access point when fine-tuning is expensive and new robot demonstrations are costly to collect.The paper motivates probing and inference-time steering as ways to use representations without modifying model weights or additionally instrumenting the environment.Source: §1 · p. 1
  • Unlike relatively static LLM concepts, VLA quantities such as object pose or contact force can depend on the action trajectory and resulting dynamics.A useful framework must address how a physical quantity evolves during task execution, not only read out its current value.Source: §1 · p. 2
  • The paper formulates VLA probing and steering around controlled physical dynamics, extending the LRH beyond static semantic concepts.The formulation addresses how action trajectories and system dynamics shape future quantities of interest.Source: §1 · p. 2
  • The paper establishes shared representations that linearly probe propagated quantities and introduces signature-based generalized linear policies for steering.Its contributions also include a control-theoretic formulation and an oracle-to-pretrained-VLA validation program.Source: §1 · p. 2

2 A CONTROL-THEORETIC FORMULATION OF VLA MODELS

The formulation treats a VLA as a policy acting on representations of observations while actions evolve an unobserved physical state. It therefore defines probing and steering in terms of action-conditioned changes to physical quantities, rather than only static readouts.

2.1 VLA MODELS AS CONTROLLED DYNAMICAL SYSTEMS

  • A VLA receives observations of an unobserved physical state and samples an open-loop action trajectory from a policy conditioned on its representation.The abstraction allows observations to include visual inputs, multiple camera views, or proprioceptive robot state.Source: §2.1 · p. 3
  • The framework treats a quantity of interest as a real-valued function of physical state, including coordinates and other physically meaningful scalars.Examples include a control Lyapunov function or value function.Source: §2.1 · p. 3

2.2 PROBING AND STEERING IN VLA MODELS

  • VLA probing and steering must account for representations being coupled to actions that induce physical state transitions.The formulation adapts both operations to this representation–action–environment interaction.Source: §2.2 · p. 3
  • Probing asks whether the representation and a candidate action trajectory suffice to predict the propagated quantity without direct access to state or dynamics.The target is the future value after the candidate trajectory, rather than only the quantity at the current state.Source: §2.2 · pp. 3–4
  • Existing VLA probing studies focus on static quantities, which do not capture action-conditioned physical evolution.The distinction matters because VLA representations both generate state-transitioning actions and provide a space in which those actions can be steered.Source: §2.2 · p. 4
  • Steering asks whether changing the representation can alter the policy's action trajectory to induce a desired change in the propagated quantity.The desired effect is on the physical outcome after the policy acts.Source: §2.2 · p. 4
  • VLA steering maps need not be simple algebraic translations; prior approaches include conceptor subspaces and learned distribution-matching maps.The formulation frames probing and steering as complementary questions about predicting and changing action-conditioned effects.Source: §2.2 · p. 4

3 LINEAR PROBING AND STEERING IN VLA MODELS

The paper defines VLA linear probing and steering for propagated physical quantities. Their linear structures depend on actions and, for steering, generally on the current state.

3 LINEAR PROBING AND STEERING IN VLA MODELS

  • Linear probing represents a propagated quantity with an action-conditioned measurement map, rather than the single global direction typical of LLM probes.The static probing setting is recovered when there is no state transition and action dependence is unnecessary.Source: §3 · p. 4
  • A key probing challenge is constructing a map from action trajectories to representation-space measurement directions because VLAs lack an analogous predefined structure to an LLM unembedding.The map must account for the candidate action trajectory.Source: §3 · p. 4
  • Linear steering uses a state-conditioned representation direction because the same action change can affect the propagated quantity differently at different states.This contrasts with standard LLM steering approaches that use a fixed global direction.Source: §3 · p. 5
  • A steering guarantee requires characterizing how linear representation translation changes the policy distribution and resulting physical behavior.The paper therefore seeks a generalized linear policy parameterization with explicit geometry, which modern generative VLA policies do not naturally provide.Source: §3 · p. 5

4 EXISTENCE OF REPRESENTATION AND MEASUREMENT MAPS

The paper constructs shared representations for multiple physical quantities by using action-path signatures to parameterize action-conditioned linear probes. Under stated reachability and regularity conditions, the construction gives an existential theoretical result, not a claim that learned VLA representations match it exactly.

4 EXISTENCE OF REPRESENTATION AND MEASUREMENT MAPS

  • The section asks whether one VLA representation can retain information for multiple quantities and support an action-conditioned linear measurement for each.These are the two questions addressed by the existence result.Source: §4 · p. 5
  • An action trajectory's signature provides an infinite-dimensional realization of the action-conditioned measurement map, with finite-dimensional maps obtained by linear transformation.The signature encodes a continuous-time path through iterated integrals and its geometric and sequential structure.Source: §4 · p. 5
  • Under uniform Chen–Fliess conditions on the horizon-H reachable set from compact K, a shared representation and QoI-specific linear maps exist for any finite collection of quantities and ε > 0.Each measurement map is the composition of a linear map with the action-path signature.Source: §4 · p. 5
  • The constructive result uses the Chen–Fliess expansion but does not assume a learned VLA representation exactly matches the Lie-derivative construction.The construction supplies an existence framework and an oracle for quantitatively assessing arbitrary learned representations.Source: §4 · p. 6
  • The signature parameterizes action-dependent measurement directions within QoI-specific subspaces of the representation space.The theorem answers the section's questions affirmatively in an existential sense.Source: §4 · p. 6

5 GEOMETRY OF THE REPRESENTATION SPACE

The policy’s signature exponential-family geometry connects VLA representations to stochastic action distributions and defines steering as a linear path in natural-parameter space. With an exact linear probe, expected future QoI is nondecreasing along that path; finite-probe-error guarantees remain open.

5 GEOMETRY OF THE REPRESENTATION SPACE

  • The signature GLM treats a VLA representation as the state-conditioned natural parameter of a distribution over action trajectories.Action paths are lifted into signature space, whose linear functionals can approximate continuous path functionals.Source: §5 · pp. 6–7
  • The signature GLM is an architecture-agnostic abstraction of how representation changes link to downstream action distributions.It supplies a statistical geometry for analyzing representation-policy interactions rather than characterizing specific VLA architectures.Source: §5 · p. 7
  • Steering is formulated as a reverse-KL projection onto a dual affine hyperplane for the desired expected QoI.Dual Bregman geometry makes the solution a linear path in natural-parameter space.Source: §5 · p. 7
  • Under an exact linear probe, expected propagated QoI is nondecreasing along the linear steering path in natural-parameter space.The result applies while the steering path remains in the natural parameter space; quantitative guarantees under finite probing error are left for future work.Source: §5 · p. 7

6 EXPERIMENTAL RESULTS

The experiments verify the proposed probing and steering structure in an oracle setting, then test it in OpenVLA-OFT. In the pretrained model, action-conditioned probes predict chunk effects and yield steering directions that change physical quantities across held-out states.

6.1 NUMERICAL VERIFICATION

  • 0.000, +0.134, and +0.255 are the expected propagated QoI values at α = 0, 4, and 8 in the oracle test.The probe is exact to 3 × 10−16 over 671 candidates; the response saturates at 0.688 as steering concentrates probability on extremal chunks.Source: Figure 2 · p. 8, §6.1 · p. 8

6.2 PROBING AND STEERING A PRETRAINED VLA

  • R² 0.90–0.98 for the bilinear increment probe predicts chunk-induced mug movement on held-out episodes, versus 0.80–0.95 for the additive probe.Static-probe R² for current mug positions is 0.64 and 0.62, versus 0.95 and 0.96 with the increment for propagated positions.Source: Table 1 · p. 9, §6.2 · p. 9
  • After six replans over 2.4 seconds, changing α shifts the mug distances while the policy continues toward the white mug.For the white mug, distance is 20/11.8/9 cm at α = −16/0/16; steering with the yellow-and-white mug probe changes that mug’s distance to 29.8/26.9/24.1 cm.Source: §6.2 · p. 9
  • 138 of 140 states show monotonic QoI increases under Dϕᵀλψ, with cosine similarity 0.98–1.00 to the simulator gradient for all seven QoIs.The direction combines the action-conditioned probe with the model’s action map and is obtained without a simulator.Source: §6.2 · p. 9, Table 2 · p. 10

7 CONCLUSION

The paper formulates linear probing and steering for evolving physical quantities in VLA models, with closed-loop steering that biases the path without changing the policy’s chosen object. Its experiments support the framework as a proof of concept, while a naturally suitable representation remains open.

7 CONCLUSION

  • The framework links action-conditioned linear probes of propagated quantities with monotonic steering directions from a signature generalized linear model.The formulation centers on how physical quantities evolve under candidate actions.Source: §7 · p. 10
  • Experiments support the framework at oracle and pretrained-VLA levels, but finding a representation naturally suited to both probing and steering remains open.The authors characterize the experiments primarily as a proof of concept for existence and realizability.Source: §7 · p. 10
  • Closed-loop steering biases the path and pace toward or away from a targeted mug while the policy still heads for the white mug, its first subgoal.Figure 3 applies steering at each re-plan over six chunks in a held-out episode.Source: Figure 3 · p. 10

POLICY ON LLM USE FOR RESEARCH AND WRITING PAPERS

The paper discloses how generative AI was used and states that its mathematical formulations were independently verified. The appendix sets out regularity conditions for a locally uniform Chen–Fliess expansion and introduces notation for its input words.

POLICY ON LLM USE FOR RESEARCH AND WRITING PAPERS

  • Generative AI assisted with language polishing and discussion or refinement of the theoretical formulation; authors independently verified all mathematical statements and final formulations.Source: POLICY ON LLM USE FOR RESEARCH AND… · p. 11

A REGULARITY CONDITIONS FOR THE CHEN–FLIESS EXPANSION

A.1 SUFFICIENT CONDITIONS FOR LOCALLY UNIFORM CONVERGENCE

Under stated reachability, regularity, and boundedness conditions, the appendix establishes absolute and uniform convergence of the Chen–Fliess expansion over compact state sets and admissible controls.

A.1 SUFFICIENT CONDITIONS FOR LOCALLY UNIFORM CONVERGENCE

  • Every initial state in the compact set must admit a unique solution over the horizon, with trajectories contained in a common compact reachable domain.Source: §A.1 · p. 15
  • The vector fields and QoIs are assumed real analytic, and admissible controls are uniformly bounded.Source: §A.1 · p. 15
  • Uniform Lie-derivative growth is bounded by constants that may depend on the compact set but not on the QoI, state, or word.Source: §A.1 · p. 15
  • The horizon and admissible control magnitude must satisfy the local convergence-radius condition qK < 1.Source: §A.1 · p. 16
  • The Chen–Fliess series converges absolutely and uniformly over the stated compact states, admissible controls, and QoIs under the sufficient conditions.The proposition states locally uniform expansion on the compact set.Source: §A.1 · p. 16

B.1 PROOF OF THEOREM 4.2

The proof constructs shared linear representations and action-conditioned measurement maps for propagated quantities, with locally uniform approximation. It also shows that the expected propagated quantity is nondecreasing along an affine steering path within the parameter domain.

B.1 PROOF OF THEOREM 4.2

  • The construction uses a shared representation and linear coordinate projections to form action-conditioned linear measurement maps for QoIs.Source: §B.1 · pp. 18–20
  • The resulting Chen–Fliess approximation holds locally uniformly in the state variable.Source: §B.1 · p. 20

B.2 PROOF OF THEOREM 5.3

  • The log-partition Hessian is the covariance operator of the sufficient statistic, making the steering-path directional derivative nonnegative.Source: §B.2 · pp. 20–21
  • Consequently, the expected propagated QoI is nondecreasing along the affine steering path when that path remains in the parameter domain.Source: §B.2 · p. 21

C DETAILED DERIVATIONS

The derivations connect KL projection to QoI-level steering and construct an oracle signature measurement map; OpenVLA-OFT experiments test action-conditioned linear probing on held-out episodes.

C.1 KL PROJECTION AND THE INDUCED STEERING PATH

  • Under a regular, minimal exponential family, the expected propagated QoI is affine in the dual coordinate, so a target level defines a constant-QoI hyperplane for reverse-KL projection.The derivation expresses KL divergence as a dual Bregman divergence before considering projection onto that hyperplane.Source: §C.1 · p. 22

C.2 ORACLE REPRESENTATION AND MEASUREMENT MAP

  • For the oracle QoI ψ(x)=py, iterated Lie derivatives terminate at order two, yielding an exact finite Chen–Fliess expansion.The oracle treats the signature measurement map Lψ as given, not learned; its representation-dimensionality scaling and possible QoI-subspace intersections require further consideration.Source: §C.2 · pp. 23–24

D.1 SETUP

  • OpenVLA-OFT's stored action-token representation reproduces its 8-step chunk, so interventions on it act directly on the action; evaluation uses held-out episodes and mug and end-effector QoIs.The study rolls out the policy from 50 initial layouts and collects snapshots for candidate-chunk evaluation.Source: §D.1 · p. 25

D.2 ACTION-CONDITIONED LINEAR PROBING

  • Theorem 3.1's propagated-QoI probe reaches R2 0.59–0.96; mug-height R2 is 0.64 and 0.62 with static probes versus 0.95 and 0.96 bilinearly, while increments score 0.90–0.98 versus 0.80–0.95 additively.Replacing the linear readout or bilinear structure with an MLP improves R2 by at most about 0.03.Source: Table 3 · p. 26, Table 4 · p. 26, §D.2 · p. 26
  • The bilinear increment captures state-dependent effects of deviations from the policy chunk; additive and bilinear probes coincide for end-effector control QoIs.The signature term captures deviations from the policy's own chunk, and the bilinear term captures how those effects depend on state.Source: §D.2 · p. 26

D.3 STEERING ALONG THE PROBE DIRECTION

The probe direction transported through the action head closely matches the simulator’s steering direction, whereas static and untransported probe directions are weaker or can steer the wrong way. In deterministic OpenVLA-OFT, this supports effective linear steering within the leading 32 principal components, subject to probe accuracy for induced chunk variations.

Evaluation

  • Steering moves only the representation supplied to the action head while holding physical state fixed; directions and probes are fitted on one episode half and evaluated on the other.Evaluation uses 20 selected states per QoI, with states chosen for QoI spread and, for mug heights and mug–plate distances, action responsiveness.Source: §D.3 · pp. 27–28

Method

  • Transporting λψ through the action-head Jacobian gives Dϕ⊤λψ, the gradient of the probe-predicted QoI with respect to representation and a direction that needs no simulator query.For deterministic OpenVLA-OFT, the stochastic exponential-tilting guarantee of Theorem 5.3 does not apply; this experiment instead supports the complementary Theorem 3.2 steering direction.Source: §D.3 · pp. 27, 28, 30
  • The transported direction guarantees a first-order increase in the probe prediction, but the true QoI slope depends on residual accuracy for the chunk variations induced by steering.High R² on the data does not imply this accuracy, and monotonicity beyond first order is not guaranteed.Source: §D.3 · p. 28

Results

  • At all 140 held-out states, the simulator gradient raised each of seven QoIs monotonically by +9 to +13 mm, establishing a linear steering direction within the leading 32 principal components.The seven QoIs include four task-level distances that are nonlinear functions of object positions.Source: §D.3 · p. 28, Table 5 · p. 29
  • 138 of 140 states were monotone under Dϕ⊤λψ, which matched the oracle effect for every QoI and had median cosine 0.98–1.00 with the simulator gradient.The measured-to-probe-predicted slope ratio had median 1.01–1.26; the two failures occurred where the probe was inaccurate for induced chunk variations.Source: §D.3 · pp. 28–29, Table 5 · p. 29, Figure 5 · p. 30
  • The static direction lowered all three heights and moved the white mug farther from its plate; its effect varied by QoI and was often opposite the simulator gradient.For the white mug–plate negative-distance QoI, the median effect was −11.3 mm, with cosine −0.61 to the simulator gradient.Source: §D.3 · pp. 29–30, Figure 5 · p. 30
  • The probe gradient γψ alone remained below Dϕ⊤λψ: it was near-orthogonal or close to random for several heights, though it reached 59–74% of oracle effect for some distances.Its effect varied by QoI; for the EE-to-yellow-and-white-mug distance, the effect was +1.7 versus +9.4 mm for the oracle.Source: §D.3 · p. 30
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