The weighted nonlinear least-squares problem for low-rank signal estimation is considered. The problem of constructing a numerical solution that is stable and fast for long time series is addressed. A modified weighted Gauss-Newton method, which can be implemented through the direct variable projection onto a space of low-rank signals, is proposed. For a weight matrix which provides the maximum likelihood estimator of the signal in the presence of autoregressive noise of order $p$ the computational cost of iterations is $O(N r^2 + N p^2 + r N \log N)$ as $N$ tends to infinity, where $N$ is the time-series length, $r$ is the rank of the approximating time series. Moreover, the proposed method can be applied to data with missing values, without increasing the computational cost. The method is compared with state-of-the-art methods based on the variable projection approach in terms of floating-point numerical stability and computational cost.

翻译：考虑了用于低级信号估计的加权非线性最小方块问题。构建一个长期稳定和快速的数字解决方案的问题得到了解决。建议采用经修改的加权高尔斯-牛顿方法,该方法可以通过直接变量投影到低级信号空间来实施。对于在自动递减噪音情况下提供信号最大可能性估测器的权重矩阵,以美元表示顺序的自动递减量 $,迭代成本的计算成本为美元(N r/2+ N p ⁇ 2 + r n\log N),以美元表示不固定,以美元表示时间序列长度为美元,以美元表示,以美元表示接近时间序列的等级为美元。此外,拟议方法可适用于缺少值的数据,但不增加计算成本。该方法与基于浮动点数字稳定性和计算成本变量预测方法的先进方法进行比较。