Intrinsic-Energy Joint Embedding Predictive Architectures Induce Quasimetric Spaces

TL;DR

This paper links joint-embedding predictive architectures with quasimetric spaces by showing that intrinsic energies in JEPAs naturally induce quasimetric structures, relevant for goal-directed control with asymmetric dynamics.

Contribution

It establishes a theoretical connection between JEPAs and quasimetric spaces through intrinsic energy functions, providing a principled basis for asymmetric control representations.

Findings

Intrinsic energies are quasimetric under mild assumptions.

Optimal cost-to-go functions can be expressed as intrinsic energies.

Symmetric energies are mismatched with one-way reachability.

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.