TY - JOUR

T1 - Low-rank signal subspace: parameterization, projection and signal estimation

AU - Zvonarev, Nikita

AU - Golyandina, Nina

N1 - Publisher Copyright: © 2023. Statistics and its Interface. All Rights Reserved.

PY - 2023

Y1 - 2023

N2 - 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.

AB - 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.

KW - Linear recurrence relation

KW - Signal estimation

KW - Signal subspace

KW - Structured low-rank approximation

KW - The gauss–newton method

KW - Time series

KW - Variable projection

UR - http://www.scopus.com/inward/record.url?scp=85135404132&partnerID=8YFLogxK

U2 - 10.4310/21-SII709

DO - 10.4310/21-SII709

M3 - Article

AN - SCOPUS:85135404132

VL - 16

SP - 117

EP - 132

JO - Statistics and its Interface

JF - Statistics and its Interface

SN - 1938-7989

IS - 1

ER -