Identifiability and data-adaptive RKHS Tikhonov regularization in nonparametric learning problems

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Johns Hopkins University
We provide an identifiability analysis for the learning problems (1) nonparametric learning of kernels in operators and (2) unsupervised learning of observation functions in state space models (SSMs). We show that in either case the function space of identifiability (FSOI) from the quadratic loss functional is the closure of a system-intrinsic data-adaptive reproducing kernel Hilbert space (SIDA-RKHS). We introduce a new method, the Data-Adaptive RKHS Tikhonov Regularization method (DARTR). The regularized estimator is robust to noise and converges as data refines. The effectiveness of DARTR is demonstrated through the following problems (1) nonparametric learning of kernels in linear/nonlinear/nonlocal operators and (2) homogenization of wave propagation in meta-material. We introduce a nonparametric generalized moment method to estimate non-invertible observation functions in nonlinear SSMs. Numerical results shows that the first two moments and temporal correlations, along with upper and lower bounds, can identify functions ranging from piecewise polynomials to smooth functions. The limitations, such as non-identifiability due to symmetry and stationary, are also discussed.
ill-posed inverse problem, identifiability, RKHS, Tikhonov regularization