Low-rank signal subspace: parameterization, projection and signal estimation

TL;DR

This paper develops a theoretical framework and algorithms for low-rank signal subspace estimation, focusing on Hankel structured problems, with new stable algorithms and an improved Gauss-Newton implementation.

Contribution

It introduces a parameterization of low-rank time series subspaces, analyzes their properties, and proposes a stable projection algorithm and an enhanced Gauss-Newton method.

Findings

The proposed projection algorithm is stable and computationally efficient.

The new Gauss-Newton implementation improves convergence and stability.

Theoretical and numerical comparisons demonstrate advantages over existing methods.

Abstract

The paper contains several theoretical results related to the weighted nonlinear least-squares problem for low-rank signal estimation, which can be considered as a Hankel structured low-rank approximation problem. A parameterization of the subspace of low-rank time series connected with generalized linear recurrence relations (GLRRs) is described and its features are investigated. It is shown how the obtained results help to describe the tangent plane, prove optimization problem features and construct stable algorithms for solving low-rank approximation problems. For the latter, a stable algorithm for constructing the projection onto a subspace of time series that satisfy a given GLRR is proposed and justified. This algorithm is used for a new implementation of the known Gauss-Newton method using the variable projection approach. The comparison by stability and computational cost is performed theoretically and with the help of an example.