Learning the dynamical system behind sensory data.

This letter looks at the physics behind sensory data by identifying the parameters that govern the physical system and estimating them from sensory observations. We extend Takens's delay-embedding theorem to the dynamical systems controlled by parameters. An embedding of the product space of the phase and the parameter spaces of the dynamical system can be obtained by the delay-embedding map, provided that the parameter of the dynamical system changes slowly. The reconstruction error is bounded for slowly varying parameters. A manifold learning technique is applied to the embedding obtained to extract a low-dimensional global coordinate system representing the product space. The phase space of the deterministic dynamics can be contracted by using the adjacency relationship in time, which enables recovery of only the parameter space. As examples, the manifolds of synthetic and real-world vowels with time-varying fundamental frequency (F(0)) are analyzed, and the F(0) contours are extracted by an unsupervised algorithm. Experimental results show that the proposed method leads to robust performance under various noise conditions and rapid changes of F(0) compared with the current state-of-the-art F(0) estimation algorithms.