The linear representation hypothesis (LRH) has become a standard lens for measuring and intervening on semantic information inside large language models (LLMs). Recent work extends this lens to vision‑language‑action (VLA) models, but the embodied, dynamical nature of interaction adds a new difficulty. Unlike static semantic attributes studied in LLMs such as gender or language, a physical quantity of interest $q_t$ in a VLA co‑evolves with system dynamics: the representation influences the policy’s action choice, the action changes the physical state, and the next representation follows.
We propose a signature‑based theoretical formulation of LRH that unifies representations and policies. On the representation side, we prove the existence of a mapping $\phi(s_t)$ such that, given a candidate action trajectory $\{a_{t:t+T}\}$, the future quantity $q_{t+T}$ can be recovered by linear probing $W\phi(s_t)$. On the policy side, we introduce a signature generalized linear model (GLM) for stochastic action chunks. Along linear paths in the natural‑parameter space, the expected future $q$ changes monotonically, enabling linear steering.
In a planar control‑affine navigation experiment we construct an explicit oracle representation and empirically confirm the predicted linear probing and steering mechanisms.
Review