Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA.
Phys Rev Lett. 2010 Feb 12;104(6):060201. doi: 10.1103/PhysRevLett.104.060201. Epub 2010 Feb 10.
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 interpolation 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 geodesic acceleration to the usual step.
将模型参数拟合到实验数据是一项常见但具有挑战性的任务，特别是如果模型包含许多参数的情况下。通常，算法会在模型对参数变化不敏感的参数空间区域中迷失方向，而需要手动进行调整。我们通过将拟合过程解释为广义插值过程来解释这种困难。通过考虑数据空间中所有模型预测的流形，我们发现截面具有层次结构的宽度，通常非常窄。当算法接近边界时，它们会卡住。我们观察到，模型流形不仅受到严格的限制，而且具有低的外曲率，导致在拟合过程中使用测地线。我们通过向通常的步骤添加测地线加速来改进 Levenberg-Marquardt 算法的收敛性。
