Projected nonlinear least squares for exponential fitting

The modern ability to collect vast quantities of data poses a challenge for parameter estimation problems. When posed as a nonlinear least squares problem fitting a model to data, the cost of each iteration grows linearly with the amount of data and it can easily become prohibitively expensive to perform many iterations. Here we develop an approach that projects the data onto a low-dimensional subspace of the high-dimensional data that preserves the information in the original data. We provide results from both optimization and statistical perspectives showing that the information is preserved when the subspace angles between this projection and the Jacobian of the model at the current iterate remain small. However, for this approach to reduce computational complexity, both the projected model and Jacobian must be computed inexpensively. This is a constraint on the pairs of models and subspaces for which this approach provides a computational speedup. Here we consider the exponential fitting problem projected onto the range of Vandermonde matrix, for which the projected model and Jacobian can be computed in closed form using a generalized geometric sum formula. We further provide an inexpensive heuristic that picks this Vandermonde matrix so that the subspace angles with the Jacobian remain small and use this heuristic to update the subspace during optimization. Although the asymptotic cost still depends on the data dimension, the overall cost of this sequence of projected nonlinear least squares problems is less expensive than the original nonlinear least squares problem. Applied to the exponential fitting problem, this provides an algorithm that is not only faster in the limit of large data than the conventional nonlinear least squares approach, but is also faster than subspace based approaches such as HSVD.