Dynamics-Adapted Cone Kernels

Dimitrios Giannakis

doi:10.1137/140954544

Abstract

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.

Highlights

Large-scale datasets generated by dynamical systems arise in a diverse range of disciplines in science and engineering, including fluid dynamics [36, 58], materials science [41, 40], molecular dynamics [23, 49], and geophysics [46, 25]
In climate atmosphere ocean science (CAOS) the dynamics take place in an infinite-dimensional phase space where the coupled nonlinear partial differential equations for fluid flow and thermodynamics are defined, and the observed data correspond to functions of that phase space, such as temperature or circulation measured over a geographical region of interest
The main theme of this paper is that kernels with suitably incorporated empirical estimates of the dynamical vector field lead to advances in both of these objectives

Summary

Introduction

Large-scale datasets generated by dynamical systems arise in a diverse range of disciplines in science and engineering, including fluid dynamics [36, 58], materials science [41, 40], molecular dynamics [23, 49], and geophysics [46, 25]. In climate atmosphere ocean science (CAOS) the dynamics take place in an infinite-dimensional phase space where the coupled nonlinear partial differential equations for fluid flow and thermodynamics are defined, and the observed data correspond to functions of that phase space, such as temperature or circulation measured over a geographical region of interest. There exists a strong need for data analysis algorithms to extract and create reduced representations of the large-scale coherent patterns which are an outcome of these dynamics, including the El Nino Southern Oscillation (ENSO) in the ocean [62] and the c 2015 SIAM. Advances in the scientific understanding and forecasting capability of these phenomena have a potentially high socioeconomic impact

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Conclusion