Design of variable densities for least-squares approximations
Abstract
We study the problem of interpolating a signal using samples at coordinates drawn for a probability density over the domain of definition of the signal, with the assumption that it can be approximated in a known linear subspace. Our goal is to minimize the number of samples needed to ensure a well-conditioned estimation of the signal. We show that the problem of optimizing the probability density is convex, and that applying the Frank-Wolf algorithm yields a simple and interpretable optimization procedure. Examples of optimizations are given with polynomials, trigonometric polynomials and Fourier-Bessel functions for wavefield interpolation.