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We consider best approximation problems in a nonlinear subset $mathcal{M}$ of a Banach space of functions $(mathcal{V},|bullet|)$. The norm is assumed to be a generalization of the $L^2$-norm for which only a weighted Monte Carlo estimate $|bullet|_n$ can be computed. The objective is to obtain an approximation $vinmathcal{M}$ of an unknown function $u in mathcal{V}$ by minimizing the empirical norm $|u-v|_n$. We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.
We consider the problem of approximating a function in general nonlinear subsets of $L^2$ when only a weighted Monte Carlo estimate of the $L^2$-norm can be computed. Of particular interest in this setting is the concept of sample complexity, the num
This work presents the windowed space-time least-squares Petrov-Galerkin method (WST-LSPG) for model reduction of nonlinear parameterized dynamical systems. WST-LSPG is a generalization of the space-time least-squares Petrov-Galerkin method (ST-LSPG)
In this work, we study the tensor ring decomposition and its associated numerical algorithms. We establish a sharp transition of algorithmic difficulty of the optimization problem as the bond dimension increases: On one hand, we show the existence of
The alternating least squares algorithm for CP and Tucker decomposition is dominated in cost by the tensor contractions necessary to set up the quadratic optimization subproblems. We introduce a novel family of algorithms that uses perturbative corre
There are plenty of applications and analysis for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting n