Geodesic acceleration and the small-curvature approximation for nonlinear least squares

It has been shown numerically that the performance of the Levenberg-Marquardt algorithm can be improved by including a second order correction known as the geodesic acceleration. In this paper we give the method a more sound theoretical foundation by deriving the geodesic acceleration correction without using differential geometry and showing that the traditional convergence proofs can be adapted to incorporate geodesic acceleration. Unlike other methods which include second derivative information, the geodesic acceleration does not attempt to improve the Gauss-Newton approximate Hessian, but rather is an extension of the small-residual approximation to cubic order. In deriving geodesic acceleration, we note that the small-residual approximation is complemented by a small-curvature approximation. This latter approximation provides a much broader justification for the Gauss-Newton approximate Hessian and Levenberg-Marquardt algorithm. In particular, it is justifiable even if the best fit residuals are large, is dependent only on the model and not on the data being fit, and is applicable for the entire course of the algorithm and not just the region near the minimum.