Koopman Invariants as Drivers of Emergent Time-Series Clustering in Joint-Embedding Predictive Architectures
This work demystifies a key behavior in self-supervised learning for time-series analysis, connecting it to dynamical systems theory to inform more robust and interpretable models.
The paper tackled the unexplained ability of Joint-Embedding Predictive Architectures (JEPAs) to cluster time-series data by dynamical regimes, showing that JEPA's predictive objective implicitly learns the invariant subspace of the Koopman operator, with validation on synthetic data demonstrating that a near-identity linear predictor constraint forces the encoder to learn these invariants.
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.