Intrinsic-Energy Joint Embedding Predictive Architectures Induce Quasimetric Spaces

Joint-Embedding Predictive Architectures (JEPAs) aim to learn representations by predicting target embeddings from context embeddings, inducing a scalar compatibility energy in a latent space. In contrast, Quasimetric Reinforcement Learning (QRL) studies goal-conditioned control through directed distance values (cost-to-go) that support reaching goals under asymmetric dynamics. In this short article, we connect these viewpoints by restricting attention to a principled class of JEPA energy functions : intrinsic (least-action) energies, defined as infima of accumulated local effort over admissible trajectories between two states. Under mild closure and additivity assumptions, any intrinsic energy is a quasimetric. In goal-reaching control, optimal cost-to-go functions admit exactly this intrinsic form ; inversely, JEPAs trained to model intrinsic energies lie in the quasimetric value class targeted by QRL. Moreover, we observe why symmetric finite energies are structurally mismatched with one-way reachability, motivating asymmetric (quasimetric) energies when directionality matters.

💡 Research Summary

This short paper establishes a rigorous theoretical bridge between Joint‑Embedding Predictive Architectures (JEPAs) and Quasimetric Reinforcement Learning (QRL). JEPAs learn representations by predicting a target embedding from a context embedding; the prediction error can be interpreted as a scalar compatibility energy E(x, y) defined over input pairs. The authors restrict attention to a principled subclass of such energies—intrinsic (least‑action) energies—which are defined as the infimum of accumulated local effort along admissible trajectories connecting two states. Formally, for a state space X, a set of admissible C¹ trajectories Γ(x→y), and a local effort density L(x, v) ≥ c·‖v‖, the intrinsic energy is

E(x, y) = inf_{γ∈Γ(x→y)} ∫₀^{T} L(γ(t), γ̇(t)) dt,

with E(x, y) = +∞ if no admissible trajectory exists.

The paper proves that any function defined in this way automatically satisfies the four axioms of a quasimetric: non‑negativity, reflexivity (E(x, x)=0), identity of indiscernibles (E(x, y)=0 ⇒ x=y), and the triangle inequality (E(x, z) ≤ E(x, y)+E(y, z)). The proof relies on the additive nature of the integral and the existence of concatenated trajectories. Moreover, the authors show that asymmetry is generic: if admissibility is directed (i.e., a forward trajectory does not guarantee a reverse one) or if the local effort is anisotropic (L(x, v) ≠ L(x, −v)), then typically E(x, y) ≠ E(y, x).

In the QRL literature, the optimal goal‑conditioned value function V*(s, g) (often negative) is precisely an intrinsic energy: it is the infimum of accumulated running costs over feasible trajectories from state s to goal g. By defining d*(s, g) = −V*(s, g), QRL obtains a quasimetric that respects directed reachability and composes over time. Hence, the class of cost‑to‑go functions targeted by QRL coincides exactly with the class of intrinsic energies.

Building on this equivalence, the authors introduce Intrinsic‑Energy JEPAs (IE‑JEPAs). An IE‑JEPA consists of an encoder f_ϕ, a target encoder \bar f_ϕ, and a predictor p_θ that outputs a scalar score. When the learned scalar can be interpreted as the intrinsic energy defined above, the resulting energy function is guaranteed to be a quasimetric. Consequently, IE‑JEPAs lie within the hypothesis space that QRL explicitly models, providing a clean conceptual link between self‑supervised representation learning and goal‑conditioned reinforcement learning.

The paper also presents a negative result: symmetric finite‑valued energies cannot encode directed reachability. If an energy E is finite exactly on reachable pairs and symmetric (E(x, y)=E(y, x)), then the reachability relation must be symmetric, contradicting many real‑world dynamics where one‑way transitions exist. This underscores the necessity of asymmetric (quasimetric) energies when directionality matters.

Finally, the authors delineate the scope of their contribution. They do not claim that all JEPAs produce quasimetric energies; the result holds only for the intrinsic‑energy subclass. No new algorithms or empirical evaluations are provided; the work is purely theoretical, aiming to clarify the structural conditions under which JEPA‑induced energies inherit quasimetric properties. The discussion points to broader applicability beyond RL, such as diffeomorphic image registration and directed embedding models for logical inference, wherever a notion of accumulated local effort over admissible transformations is natural. Future work could involve implementing IE‑JEPAs in concrete domains, designing regularizers that enforce quasimetric constraints, and exploring how the least‑action perspective can guide the design of more principled self‑supervised and reinforcement learning systems.