Why are nonlinear fits so challenging?

TL;DR

Fitting complex models to data is difficult due to narrow, bounded manifolds in data space, causing algorithms to get stuck, but incorporating geodesic acceleration improves convergence.

Contribution

This paper introduces a geometric interpretation of the fitting process and enhances the Levenberg-Marquardt algorithm with geodesic acceleration for better convergence.

Findings

Geometric interpretation reveals narrow, bounded model manifolds in data space.

Adding geodesic acceleration improves algorithm convergence.

Fitting challenges are linked to low extrinsic curvature of the model manifold.

Abstract

Fitting model parameters to experimental data is a common yet often challenging task, especially if the model contains many parameters. Typically, algorithms get lost in regions of parameter space in which the model is unresponsive to changes in parameters, and one is left to make adjustments by hand. We explain this difficulty by interpreting the fitting process as a generalized interpretation procedure. By considering the manifold of all model predictions in data space, we find that cross sections have a hierarchy of widths and are typically very narrow. Algorithms become stuck as they move near the boundaries. We observe that the model manifold, in addition to being tightly bounded, has low extrinsic curvature, leading to the use of geodesics in the fitting process. We improve the convergence of the Levenberg-Marquardt algorithm by adding the geodesic acceleration to the usual Levenberg-Marquardt step.