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.

中文翻译：

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低秩信号子空间：参数化、投影和信号估计

该论文包含了几个与用于低秩信号估计的加权非线性最小二乘问题相关的理论结果，可以将其视为 Hankel 结构的低秩逼近问题。描述了与广义线性递归关系（GLRR）相关的低秩时间序列子空间的参数化，并研究了其特征。展示了获得的结果如何帮助描述切平面，证明优化问题的特征并构建解决低秩逼近问题的稳定算法。对于后者，提出并证明了一种将投影构造到满足给定 GLRR 的时间序列子空间上的稳定算法。该算法用于使用可变投影方法的已知高斯-牛顿方法的新实现。