Convergence bounds for empirical nonlinear least-squares

Abstract

We consider best approximation problems in a nonlinear subset M of a Banach space of functions (V, • ). The norm is assumed to be a generalization of the L 2 -norm for which only a weighted Monte Carlo estimate • n can be computed. The objective is to obtain an approximation v ∈ M of an unknown function u ∈ V by minimizing the empirical norm u − v n . In the case of linear subspaces 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 = dim(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 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.

A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression

“…We discuss the problem of function identification from data for tensor train based ansatz spaces and give some insights into when these ansatz spaces can be used efficiently. For this we combine recent results on sample complexity [EST20] and block sparsity of tensor train networks [BGP21] to motivate a novel algorithm for the problem at hand. We then demonstrate the applicability of this algorithm to different problems.…”

“…The quality of the solution u W,M of ( 14) in relation to u W is subject to tremendous interest on the part of the mathematics community. Two particular papers that consider this problem are [CM17] and [EST20]. While the former provides sharper error bounds for the case of linear ansatz spaces the latter generalizes the work and is applicable to tensor network spaces.…”

“…The bound for the dictionary of monomials Ψ monomial is more involved but can theoretically be computed in the same way. By drawing samples from an adapted sampling measure [CM17] the theory in [EST20] ensures that K(V ) = dim(V ) for all linear spaces V -independent of the underlying dictionary Ψ. Using such an optimally weighted least-squares method thus leads to the bounds…”

“…In [EST20] an important observation was made. It was shown that for tensor networks the sample complexity, meaning the number of data points needed, is related to the dimension of the high-dimensional ansatz space.…”

Dynamical low-rank approximations of solutions to the Hamilton-Jacobi-Bellman equation

“…DLR approximations to parabolic PDEs have been studied in [Bac+21;Con20], but -to the best of our knowledge -this work is the first application of DLR methods to a finite horizon optimal control problem and in particular to the nonlinear HJB equation. In order to derive an abstract DLR problem on the TT manifold, we use a Variational Monte Carlo (VMC) approach [Bay+21;EST20]. In our setting it can be understood as an empirical least squares tensor regression based on random samples.…”