Unwinding the model manifold: choosing similarity measures to remove local minima in sloppy dynamical systems

We consider the problem of parameter sensitivity in models of complex dynamical systems through the lens of information geometry. In most cases, models are sloppy, that is, exhibit an exponential hierarchy of parameter sensitivities. We propose a parameter classification based on how sensitivities scale at long observation times. We show that for oscillatory models, sensitivities can become arbitrarily large, which implies a high effective-dimensionality on the model manifold. This translates to multimodal fitting problems and stands in contrast to the low effective-dimensionality previously observed in sloppy models with a single fixed point. We define a measure of curvature on the model manifold which we call the winding frequency that estimates the density of local minima in the model's parameter space. We then show how alternative choices of fitting metrics can "unwind" the model manifold and give low winding frequencies. This prescription translates the model manifold from one of high effective-dimensionality into the "hyper-ribbon" structures observed elsewhere. This translation opens the door for applications of sloppy model analysis and model reduction methods developed for models with low effective-dimensionality.