Topological Linear System Identification via Moderate Deviations Theory

Two dynamical systems are topologically equivalent when their phase-portraits can be morphed into each other by a homeomorphic coordinate transformation on the state space. The induced equivalence classes capture qualitative properties such as stability or oscillatory nature of trajectories, for example. In this paper we develop method to learn topological class an unknown stable system from single trajectory finitely many observations. Using moderate deviations principle least squares estimator matrix $\theta$, prove that probability misclassification decays exponentially with number observations at rate is proportional square smallest singular value $\theta$.