The paper is devoted to the solution of a weighted nonlinear least-squares problem for low-rank signal estimation, which is related to Hankel structured low-rank approximation problems. A modified weighted Gauss-Newton method, which uses projecting on the image space of the signal, is proposed to solve this problem. The advantage of the proposed method is the possibility of its numerically stable and fast implementation. For a weight matrix, which corresponds to an autoregressive process of order $p$, the computational cost of iterations is $O(N r^2 + N p^2 + r N \log N)$, where $N$ is the time series length, $r$ is the rank of the approximating time series. For developing the method, some useful properties of the space of time series of rank $r$ are studied. The method is compared with state-of-the-art methods based on the variable projection approach in terms of numerical stability, accuracy and computational cost.

翻译：本文致力于解决用于低级信号估计的加权非线性最低方的问题,这个问题与Hankel结构化低级近似问题有关。建议采用使用信号图像空间投影的经修改的加权高斯-牛顿方法解决这一问题。拟议方法的优势在于其数字稳定、快速实施的可能性。对于相当于自动递减顺序美元过程的加权矩阵,迭代的计算成本是$O(N r2+N p ⁇ 2+ r n\log N)$,其中,美元是时间序列的长度,美元是大约时间序列的等级。为开发方法,研究了美元级时间序列空间的一些有用属性。该方法与基于数字稳定性、准确性和计算成本可变预测方法的先进方法进行了比较。