Source-linked AI summary
Why are nonlinear fits so challenging?
M. K. Transtrum, B. B. Machta, J. P. Sethna
TL;DR
The paper interprets difficult multiparameter fitting geometrically, focusing on narrow model-manifold boundaries and low extrinsic curvature. It adds geodesic acceleration to the Levenberg-Marquardt step to improve convergence with a more accurate trust region and reduced computation time.
Problem
Narrow boundaries on the model manifold explain why fitting algorithms can become stuck.
Method
The paper treats multiparameter models as a high-dimensional analytic interpolation scheme and uses geodesics in the fitting process.
Results
Geodesic acceleration improves Levenberg-Marquardt convergence by providing a more accurate trust region while reducing computation time.
Takeaways & Limitations
The model-manifold geometry motivates augmenting the usual Levenberg-Marquardt step with geodesic acceleration.
Abstract
from arXiv · showhide
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.
R < 1 b y the ratio test. Ea h ross se -
Nonlinear fits are difficult because model manifolds have narrow, hierarchical cross-sectional widths and low extrinsic curvature. A geometric interpretation links sloppy Hessian spectra to hyper-ribbon structure and motivates geodesic-based fitting methods.
- Manifold geometry: Many model parameters per effective degree of freedom yield near-degenerate Hessians, correlated data points, and few effective degrees of freedom.This explains why multiparameter models commonly exhibit sloppiness and hierarchical widths.
- Geodesic coordinates: Geodesic coordinates make cost contours near the best fit nearly quadratic and isotropic, despite the original nonlinear parameterization.The paper uses geodesics to construct polar coordinates on the model manifold.
- Algorithmic implication: The model graph stretches manifold boundaries toward infinity in parameter space, providing a geometric basis for modifying Levenberg-Marquardt fitting.The graph combines the model manifold embedded in data space with parameter space and uses an interpolated metric.
- Manifold geometry: Model-manifold widths form a hierarchy that mirrors the hierarchy of Hessian eigenvalues, producing the hyper-ribbon structure characteristic of sloppy nonlinear models.The widths and singular values each span around four orders of magnitude, while the curvatures span eight.
- Curvature: The model manifold typically has low extrinsic curvature, while parameter-effects curvature is much larger, making geodesics nearly straight in data space.The extrinsic curvatures are reported as three orders of magnitude smaller than the parameter-effects curvature.