Error Guarantees for Least Squares Approximation with Noisy Samples in Domain Adaptation

Given $n$ samples of a function $f\colon D\to\mathbb C$ 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 $Vm$, $\dim(Vm) = m < \infty$ 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 $H{\mathrm{mix}}^2$. Overcoming numerical issues of this $H_{\text{mix}}^2$ basis, this gives a novel stable approximation method with quadratic error decay. Numerical experiments indicate the applicability of our results.