Estimating linear response statistics using orthogonal polynomials: An RKHS formulation

Abstract

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.

arXiv comment31 pages, 17 figures

References in corpus (10)

• Fluctuations and response of nonequilibrium states
• A simple framework to justify linear response theory
• The computation of averages from equilibrium and nonequilibrium Langevin molecular dynamics
• Density Estimation in Infinite Dimensional Exponential Families
• Limitations of polynomial chaos expansions in the Bayesian solution of inverse problems
• Generalization Error Bounds of Gradient Descent for Learning Over-parameterized Deep ReLU Networks
• Thermodynamic formalism and linear response theory for non-equilibrium steady states
• A Parameter Estimation Method Using Linear Response Statistics
• The strong convergence of operator-splitting methods for the Langevin dynamics model
• Eigenvalues of the Laplacian on a compact manifold with density

Cited by in corpus (1)

• Linear Response Based Parameter Estimation in the Presence of Model Error