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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 that 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.

Fast and stable modification of the Gauss–Newton method for low‐rank signal estimation