Convergence bounds for empirical nonlinear least-squares

We consider best approximation problems in a nonlinear subset of a Banach space of functions. The norm is assumed to be a generalization of the L2 norm for which only a weighted Monte Carlo estimate can be computed. The objective is to obtain an approximation of an unknown target function by minimizing the empirical norm. In the case of linear subspaces it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples is too close to the number of parameters. We review this statement 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 we show sufficient conditions for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds

Convergence bounds for empirical nonlinear least-squares

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 $v\in\mathcal{M}$ of an unknown function $u \in \mathcal{V}$ by minimizing the empirical norm $\|u-v\|_n$. In the case of linear subspaces $\mathcal{M}$ it is well-known that such least squares approximations can become inaccurate and unstable when the number of samples $n$ is too close to the number of parameters $m = \operatorname{dim}(\mathcal{M})$. We review this statement 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 we show that $n \gtrsim m$ is sufficient for the RIP to be satisfied with high probability. Several model classes are examined where analytical statements can be made about the RIP. Numerical experiments illustrate some of the obtained stability bounds.Comment: 32 pages, 18 figures; major revision