Henry Brown, Temple University
Abstract: We study the problem of noisy data fitting and spectral measure recovery in the classes Stieltjes and Completely monotone functions. The analytical setting is non-negative least squares over the conical hull of a curve in Rn. While the non-negative least squares problem receives much attention, no method exists to analyze convergence to the true solution, and popular numerical algorithms come with very few guarantees. We remedy this by developing a simple theory of minimization over convex cones, by which we can convert the non-negative constraints to convex ones. In this context, convergence analysis is well understood. Algorithms for the convex problem provide means to compute a function along with its spectral measure which provide a good fit for the noisy data and approximately satisfy the optimality conditions. Our theory then gives us tools to analyze the support of the optimal spectral measure.