TY - JOUR

T1 - ESTIMATING LINEAR RESPONSE STATISTICS USING ORTHOGONAL POLYNOMIALS

T2 - AN RKHS FORMULATION

AU - Zhang, He

AU - Harlim, John

AU - Li, Xiantao

N1 - Funding Information:
33C50, 37A25. Key words and phrases. Linear response theory, kernel embedding, orthogonal polynomial, reproducing kernel hilbert space, Mercer’s theorem, Mercer-type kernel. The research of XL was supported under the NSF grant DMS-1819011 and JH was supported under the NSF grant DMS-1854299. ∗ Corresponding author.
Publisher Copyright:
© American Institute of Mathematical Sciences.

PY - 2020/12

Y1 - 2020/12

N2 - We study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. Based on the fluctuation-dissipation theory, this problem is reformulated as an unsupervised learning task of estimating a density function. We consider a nonparametric density estimator formulated by the kernel embedding of distributions with “Mercer-type” kernels, constructed based on the classical orthogonal polynomials defined on non-compact domains. While the resulting representation is analogous to Polynomial Chaos Expansion (PCE), the connection to the reproducing kernel Hilbert space (RKHS) theory allows one to establish the uniform convergence of the estimator and to systematically address a practical question of identifying the PCE basis for a consistent estimation. We also provide practical conditions for the well-posedness of not only the estimator but also of the underlying response statistics. Finally, we provide a statistical error bound for the density estimation that accounts for the Monte-Carlo averaging over non-i.i.d time series and the biases due to a finite basis truncation. This error bound provides a means to understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernels. Numerically, we verify the effectiveness of the estimator on two stochastic dynamics with known, yet, non-trivial equilibrium densities.

AB - We study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. Based on the fluctuation-dissipation theory, this problem is reformulated as an unsupervised learning task of estimating a density function. We consider a nonparametric density estimator formulated by the kernel embedding of distributions with “Mercer-type” kernels, constructed based on the classical orthogonal polynomials defined on non-compact domains. While the resulting representation is analogous to Polynomial Chaos Expansion (PCE), the connection to the reproducing kernel Hilbert space (RKHS) theory allows one to establish the uniform convergence of the estimator and to systematically address a practical question of identifying the PCE basis for a consistent estimation. We also provide practical conditions for the well-posedness of not only the estimator but also of the underlying response statistics. Finally, we provide a statistical error bound for the density estimation that accounts for the Monte-Carlo averaging over non-i.i.d time series and the biases due to a finite basis truncation. This error bound provides a means to understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernels. Numerically, we verify the effectiveness of the estimator on two stochastic dynamics with known, yet, non-trivial equilibrium densities.

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U2 - 10.3934/fods.2020021

DO - 10.3934/fods.2020021

M3 - Article

AN - SCOPUS:85100384976

SN - 2639-8001

VL - 2

SP - 443

EP - 485

JO - Foundations of Data Science

JF - Foundations of Data Science

IS - 4

ER -