Enhancing Prony's method by nuclear norm penalization and extension to missing data

Prony's method is a widely used method for modelling signals using a finite sum of exponential terms. It has innumerable applications in weather modelling, finance, medical signal analysis, image compression, time series analysis, power grids, etc. Prony's method has, however, the reputation of being unstable with respect to noise perturbations. The goal of the present paper is to assess the potential improvements of a nuclear-norm-penalized regularization of Prony's method. The nuclear norm regularization is a standard technique for improving the performance when processing noisy signals with low-rank underlying structure such as in matrix completion, matrix compressed sensing, hidden variable models. We consider both the standard setting and the case of missing data. We provide a fast estimation algorithm for the nuclear-norm-penalized least-squares minimization. Monte Carlo experiments show that regularization can significantly enhance the performance of Prony's method with and without missing data.