Koopman Invariants as Drivers of Emergent Time-Series Clustering in Joint-Embedding Predictive Architectures

TL;DR

This paper explains how Joint-Embedding Predictive Architectures (JEPAs) naturally cluster time-series data by learning Koopman eigenfunctions, linking self-supervised learning with dynamical systems theory.

Contribution

It introduces a theoretical framework showing JEPAs implicitly learn Koopman invariants, validated on synthetic data, and highlights the importance of a near-identity predictor constraint.

Findings

JEPAs learn Koopman eigenfunctions as invariants.

Constraining the linear predictor guides the encoder to meaningful invariants.

The predictor's role is crucial for disentangled, interpretable representations.

Abstract

Joint-Embedding Predictive Architectures (JEPAs), a powerful class of self-supervised models, exhibit an unexplained ability to cluster time-series data by their underlying dynamical regimes. We propose a novel theoretical explanation for this phenomenon, hypothesizing that JEPA's predictive objective implicitly drives it to learn the invariant subspace of the system's Koopman operator. We prove that an idealized JEPA loss is minimized when the encoder represents the system's regime indicator functions, which are Koopman eigenfunctions. This theory was validated on synthetic data with known dynamics, demonstrating that constraining the JEPA's linear predictor to be a near-identity operator is the key inductive bias that forces the encoder to learn these invariants. We further discuss that this constraint is critical for selecting this interpretable solution from a class of mathematically equivalent but entangled optima, revealing the predictor's role in representation disentanglement. This work demystifies a key behavior of JEPAs, provides a principled connection between modern self-supervised learning and dynamical systems theory, and informs the design of more robust and interpretable time-series models.