Henry Brown: On Recovering Optimal Spectral Measure from Noisy Data

We study the problem of noisy data fitting and spectral measure recovery in the classes Stieltjes and Completely monotone functions. The analytical setting is least squares over the convex or conical hull of a space curve in IR^n. I will begin with the case of least squares over convex hulls where the convergence analysis is simple. Here, I will introduce our method of analyzing the support of the optimal spectral measure. I will then move to the case of conical hulls (non-negative least squares). In this context, convergence analysis is not well understood. We introduce a simple assumption on the problem which allows us to overcome the convergence problem. While our assumption does not initially hold in either our motivating examples, I will show they can be rescaled so that is does hold. This rescaling allows certain algorithms for convex minimization to be extended to minimization problems with non-negativity constraints. I will provide several numerical examples which show convergence of algorithms as well as highlight our spectral measure analysis at certain noise levels.