Convergence bounds for empirical nonlinear least-squares
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We consider best approximation problems in a (nonlinear) subspace M of a Banach space (V,∙) where only an empirical estimate ∙_n of the norm can be computed. The norm is assumed to be of the form v := E_Y[|v|_Y^2]^1/2 for some (parametric) seminorm |∙|_Y depending on a random variable Y. The objective is to approximate an unknown function u ∈V by v∈M by minimizing the empirical norm u-v_n^2 := 1n∑_i=1^n |u-v|_y_i^2 w.r.t. n random samples {y_i}_i=1,...,n. 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 ∝dim(M). We review this statement in the light of adapted distributions for the samples y_i and establish error bounds of the empirical best approximation error based on a restricted isometry property (RIP) (1-δ)v^2 <v_n^2 < (1+δ)v^2 ∀ v∈M which holds in probability. These results are closely related to those in "Optimal weighted least-squares methods" (A. Cohen and G. Migliorati, 2016) and show that n > sm is sufficient for the RIP to be satisfied with high probability. The factor s represents the variation of the empirical norm ∙_n on M. choice of the distribution of the samples. Several model classes are examined and numerical experiments illustrate some of the obtained stability bounds.
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