A sequential majorization method for approximating weighted time series of finite rank

摘要

The low-rank Hankel matrix optimization has become one of the main approaches to the signal extraction from noisy time series of finite rank. The approach is particularly effective if different weights are enforced to the data points to reflect their relative importance. Two guiding principles for developing such an approach are (i) the Hankel matrix optimization should be computationally tractable, and (ii) the objective in the optimization should be a close approximation to the original weighted least-squares. In this paper, we introduce a sequential approximation that satisfies (i) and (ii) based on the technique of majorization. A new approximation is constructed as soon as a new iterate is computed from the previous approximation and it makes use of the latest gradient information of the objective, leading to more accurate an approximation to the objective. The resulting subproblem bears a similar structure to an existing scheme and hence can be efficiently solved. Convergence of the sequential majorization method (SMM) is guaranteed provided that the solution of the subproblem satisfies a sandwich inequality. We also compare SMM with two leading methods in literature on real-life problems. Significant improvement is observed in some cases.