Statistics and Its Interface

Volume 16 (2023)

Number 1

Special issue on recent developments in complex time series analysis – Part I

Guest editors: Robert T. Krafty (Emory Univ.), Guodong Li (Univ. of Hong Kong), Anatoly Zhigljavsky (Cardiff Univ.)

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

Pages: 117 – 132

DOI: https://dx.doi.org/10.4310/21-SII709

Authors

Nikita Zvonarev (Department of Statistical Modeling, St. Petersburg State University, St. Petersburg, Russia)

Nina Golyandina (Department of Statistical Modeling, St. Petersburg State University, St. Petersburg, Russia)

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 utilized 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.

Keywords

signal subspace, signal estimation, structured low-rank approximation, linear recurrence relation, time series, Gauss–Newton method, variable projection

2010 Mathematics Subject Classification

Primary 37M10, 65F30. Secondary 49M15, 94A12.

Received 21 February 2021

Accepted 17 November 2021

Published 28 December 2022