2022/04/09 by Bartel, Felix · 1 citation
#41A10 #41A25 #41A60 #41A63 #42C10 #65D15 #65F22 #65Txx #95A20 #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2204.04436
Given n samples of a function f\colon D→\mathbb C in random points drawn with respect to a measure \varrhoS we develop theoretical analysis of the L2(D, \varrhoT)-approximation error. For a parituclar choice of \varrhoS depending on \varrhoT, it is known that the weighted least squares method from finite dimensional function spaces Vm, dim(Vm) = m < ∞ has the same error as the best approximation in Vm up to a multiplicative constant when given exact samples with logarithmic oversampling. If the source measure \varrhoS and the target measure \varrhoT differ we are in the domain adaptation setting, a subfield of transfer learning. We model the resulting deterioration of the error in our bounds. Further, for noisy samples, our bounds describe the bias-variance trade off depending on the dimension m of the approximation space Vm. All results hold with high probability. For demonstration, we consider functions defined on the d-dimensional cube given in unifom random samples. We analyze polynomials, the half-period cosine, and a bounded orthonormal basis of the non-periodic Sobolev space Hmix2. Overcoming numerical issues of this Hmix2 basis, this gives a novel stable approximation method with quadratic error decay. Numerical experiments indicate the applicability of our results.