Chebyshev Approximation and the Global Geometry of Model Predictions

• Published in: Physical review letters 2019-04, Vol.122 (15), p.158302-158302, Article 158302
• Main Authors: Quinn, Katherine N, Wilber, Heather, Townsend, Alex, Sethna, James P
• Format: Article
• Language: English
• Subjects: Approximation; Chebyshev approximation; Chemical reactions; Conditioning; Epidemiology; Global geometry; Mathematical models; Organic chemistry; Parameters; Smoothness
• ISSN: 0031-9007; 1079-7114
• DOI: 10.1103/PhysRevLett.122.158302

Complex nonlinear models are typically ill conditioned or sloppy; their predictions are significantly affected by only a small subset of parameter combinations, and parameters are difficult to reconstruct from model behavior. Despite forming an important universality class and arising frequently in practice when performing a nonlinear fit to data, formal and systematic explanations of sloppiness are lacking. By unifying geometric interpretations of sloppiness with Chebyshev approximation theory, we rigorously explain sloppiness as a consequence of model smoothness. Our approach results in universal bounds on model predictions for classes of smooth models, capturing global geometric features that are intrinsic to their model manifolds, and characterizing a universality class of models. We illustrate this universality using three models from disparate fields (physics, chemistry, biology): exponential curves, reaction rates from an enzyme-catalyzed chemical reaction, and an epidemiology model of an infected population.