Kernel Embedding Linear Response
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In the paper, we study the problem of estimating linear response statistics under external perturbations using time series of unperturbed dynamics. A standard approach to this estimation problem is to employ the Fluctuation-Dissipation Theory, which, in turn, requires the knowledge of the functional form of the underlying unperturbed density that is not available in general. To overcome this issue, we consider a nonparametric density estimator formulated by the kernel embedding of distribution. To avoid the computational expense arises using radial type kernels, we consider the "Mercer-type" kernels constructed based on the classical orthogonal bases defined on non-compact domains, such as the Hermite and Laguerre polynomials. We establish the uniform convergence of the estimator, which justifies the use of the estimator for interpolation. Given a target function with a specific decaying property quantified by the available data, our framework allows one to choose the appropriate hypothesis space (an RKHS) that is "rich" enough for consistent estimation. For the linear response estimation, our study provides practical conditions for the well-posedness of both the estimator and the underlying response statistics. We offer a theoretical guarantee for the convergence of the estimator to the underlying actual linear 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 biases due to finite basis truncation. This error bound provides a mean to understand the feasibility as well as limitation of the kernel embedding with Mercer-type kernel. Numerically, we verify the effectiveness of the kernel embedding linear response estimator on two stochastic dynamics with known, yet, non-trivial equilibrium densities.
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