Worst-case Recovery Guarantees for Least Squares Approximation Using Random Samples

Abstract

We construct a least squares approximation method for the recovery of complex-valued functions from a reproducing kernel Hilbert space on $$D \subset \mathbb {R}^d$$ D ⊂ R d . The nodes are drawn at random for the whole class of functions, and the error is measured in $$L_2(D,\varrho _{D})$$ L 2 ( D , ϱ D ) . We prove worst-case recovery guarantees by explicitly controlling all the involved constants. This leads to new preasymptotic recovery bounds with high probability for the error of hyperbolic Fourier regression on multivariate data. In addition, we further investigate its counterpart hyperbolic wavelet regression also based on least squares to recover non-periodic functions from random samples. Finally, we reconsider the analysis of a cubature method based on plain random points with optimal weights and reveal near-optimal worst-case error bounds with high probability. It turns out that this simple method can compete with the quasi-Monte Carlo methods in the literature which are based on lattices and digital nets.

Sparse tensor product approximation for a class of generalized method of moments estimators

“…This way, the rather slow rate of MC and QMC quadrature can be improved significantly for functions in H ri mix (Ω i ). As yet, at least the upper bound O(N −ri i,li log(N i,li ) ridi ) for r i > 1/2 has been proven for the optimal weights MC case [32], whereas the lower bound O(N −ri i,li log(N i,li ) (di−1)/2 ) is known for general best weighted sampling. This is a major improvement w.r.t.…”

Interpretable Transformed ANOVA Approximation on the Example of the Prevention of Forest Fires

“…In our case, ω is the density of the standard normal distribution, i.e., the nodes X have to be distributed accordingly. For a detailed discussion on the properties of those matrices we refer to [26,27] where our basis is a special case.…”

Constructive subsampling of finite frames with applications in optimal function recovery

“…The associated left and right singular functions shall be denoted by (η k ) ∞ k=1 and (e k ) ∞ k=1 (see Appendix C for more details). We follow the course of [13,12,18,20], where this setting was considered as well.…”

“…We hence have K(x, x) = k∈N |e k (x)| 2 for ν-almost every x ∈ D. This is a crucial ingredient in the proof of Theorem 6.7, see also [18,Sec. 4] and [12,Sec. 2].…”

“…A related scenario is investigated in the recent papers [13,12,18,20]. Here one is interested in the recovery of functions from a reproducing kernel Hilbert space (RKHS) H(K) in L 2 (D, ν), where ν is allowed to be infinite.…”