Intrinsic-Energy Joint Embedding Predictive Architectures Induce Quasimetric Spaces
Anthony Kobanda, Waris Radji
TL;DR
The paper investigates when a JEPA-derived energy can serve as a cost-to-go in goal-conditioned control. By restricting JEPA energies to intrinsic (least-action) functionals defined as $E(x,y)=\inf_{\gamma\in\Gamma(x\to y)} \mathsf{Act}(\gamma)$, the authors show that the resulting energy satisfies the triangle inequality and becomes a quasimetric, enabling directed reachability. In this regime, intrinsic-energy JEPAs fall into the quasimetric value-function class targeted by Quasimetric Reinforcement Learning (QRL), establishing a structural link between JEPA world-model energies and QRL geometry. The note also explains why symmetric finite energies cannot encode one-way reachability and discusses scope beyond RL, including applications to deformation distances and directed implications.
Abstract
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.