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We consider the problem of approximating a function in general nonlinearsubsets of $L^2$ when only a weighted Monte Carlo estimate of the $L^2$-normcan be computed. Of particular interest in this setting is the concept ofsample complexity, the number of samples that are necessary to recover the bestapproximation. Bounds for this quantity have been derived in a previous workand depend primarily on the model class and are not influenced positively bythe regularity of the sought function. This result however is only a worst-casebound and is not able to explain the remarkable performance of iterative hardthresholding algorithms that is observed in practice. We reexamine the resultsof the previous paper and derive a new bound that is able to utilize theregularity of the sought function. A critical analysis of our results allows usto derive a sample efficient algorithm for the model set of low-rank tensors.The viability of this algorithm is demonstrated by recovering quantities ofinterest for a classical high-dimensional random partial differential equation.

Convergence bounds for nonlinear least squares and applications to tensor recovery