Topology-driven classification of time series

Time series analysis is fundamentally limited by the lack of representations that reflect the underlying generative mechanisms of observed signals. Existing approaches, ranging from spectral decompositions to modern machine learning, primarily operate on signal values or frequency content, and therefore fail to capture the intrinsic structure of the dynamics that produce the data.

In this work, we introduce a geometric framework that establishes a direct correspondence between the generative structure of a time series and the topology of its delay embedding. We show that broad classes of signals (including exponential, harmonic, and exponentially modulated oscillatory processes) induce invariant low-dimensional subspaces in Hankel embedding space, which dimension is determined solely by the number and type of latent dynamical components.

This leads to a unifying principle: the intrinsic dimension and geometry of delay embeddings act as invariants of the underlying dynamics. Building on this result, we reformulate time series classification as the problem of separating equivalence classes defined by ε -neighborhoods of subspaces on a Grassmann manifold. This yields a topological classifier that is interpretable, data-efficient, and provably robust, where noise admits a natural geometric interpretation as bounded perturbations of subspaces.

We demonstrate that the proposed framework distinguishes signals with indistinguishable spectral signatures and consistently recovers the latent structure of complex, noisy, multi-component processes. On benchmark EEG data, the method achieves state-of-the-art performance without feature engineering or large-scale training.

These results suggest a shift from feature-based and statistical representations toward a geometric theory of time series, in which structure, classification are governed by the topology of embeddings.

An interactive web-based demonstration is available to facilitate exploration of the geometric structure of delay embeddings and the proposed classification approach.