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Why are nonlinear fits to data so challenging?

Mark K Transtrum1, Benjamin B Machta, James P Sethna

  • 1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA. mkt26@cornell.edu

Physical Review Letters
|April 7, 2010
PubMed
Summary
This summary is machine-generated.

This study explains why fitting many model parameters to experimental data is difficult. By adding geodesic acceleration, the Levenberg-Marquardt algorithm

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Area of Science:

  • Computational physics
  • Data science
  • Mathematical modeling

Background:

  • Fitting model parameters to experimental data is a complex task, particularly with numerous parameters.
  • Current algorithms often struggle in parameter space regions where the model is insensitive to parameter changes, necessitating manual adjustments.

Purpose of the Study:

  • To elucidate the underlying reasons for difficulties in fitting complex models to experimental data.
  • To propose an enhancement to existing fitting algorithms to improve convergence and accuracy.

Main Methods:

  • Interpreting the model fitting process as a generalized interpolation.
  • Analyzing the geometry of the model prediction manifold in data space, including its cross-sectional widths and extrinsic curvature.
  • Modifying the Levenberg-Marquardt algorithm by incorporating geodesic acceleration.

Main Results:

  • Model prediction manifolds are characterized by a hierarchy of narrow cross sections, causing algorithms to become trapped near boundaries.
  • The model manifold exhibits low extrinsic curvature, suggesting the utility of geodesic paths.
  • The enhanced Levenberg-Marquardt algorithm demonstrates improved convergence through the addition of geodesic acceleration.

Conclusions:

  • The geometric properties of the model manifold explain convergence issues in parameter fitting.
  • Geodesic acceleration offers a viable strategy to enhance the performance of standard fitting algorithms like Levenberg-Marquardt.