Dynamics-Adapted Cone Kernels

We present a family of kernels for analysis of data generated by dynamical systems. These so-called cone kernels feature a 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. The outcome of this explicit dependence on the dynamics is that, in a suitable asymptotic limit, Laplace-Beltrami operators for data analysis constructed from cone kernels generate diffusions along the integral curves of the dynamical vector field. This property is independent of the observation modality, and endows these operators with invariance under a weakly restrictive class of transformations of the data (including conformal transformations), while it also enhances their capability to extract intrinsic dynamical timescales via eigenfunctions. Here, we study these features by establishing the Riemannian metric tensor induced by cone kernels in the limit of large data. We find that the corresponding Dirichlet energy is governed by the directional derivative of functions along the dynamical vector field, giving rise to a measure of roughness of functions that favors slowly varying observables. 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.