Dynamics-Adapted Cone Kernels

Abstract: We present a family of kernels for analysis of data generated by dynamical systems. These so-called cone kernels feature an explicit dependence on the dynamical vector field operating in the phase-space manifold, estimated empirically through finite-differences of time-ordered data samples. In particular, cone kernels assign strong affinity to pairs of samples whose relative displacement vector lies within a narrow cone aligned with the dynamical vector field. As a result, in a suitable asymptotic limit, the associated diffusion operator generates diffusions along the dynamical flow, and is invariant under a weakly restrictive class of transformations of the data, which includes conformal transformations. Moreover, the corresponding Dirichlet form is governed by the directional derivative of functions along the dynamical vector field. The latter feature is metric-independent. The diffusion eigenfunctions obtained via cone kernels are therefore adapted to the dynamics in that they vary predominantly in directions transverse to the flow. We demonstrate the utility of cone kernels in nonlinear flows on the 2-torus and North Pacific sea surface temperature data generated by a comprehensive climate model.