Parsimonious Representation of Nonlinear Dynamical Systems Through Manifold Learning: A Chemotaxis Case Study
Nonlinear manifold learning algorithms, such as diffusion maps, have been fruitfully applied in recent years to the analysis of large and complex data sets. However, such algorithms still encounter challenges when faced with real data. One such challenge is the existence of "repeated eigendirections," which obscures the detection of the true dimensionality of the underlying manifold and arises when several embedding coordinates parametrize the same direction in the intrinsic geometry of the data set. We propose an algorithm, based on local linear regression, to automatically detect coordinates corresponding to repeated eigendirections. We construct a more parsimonious embedding using only the eigenvectors corresponding to unique eigendirections, and we show that this reduced diffusion maps embedding induces a metric which is equivalent to the standard diffusion distance. We first demonstrate the utility and flexibility of our approach on synthetic data sets. We then apply our algorithm to data collected from a stochastic model of cellular chemotaxis, where our approach for factoring out repeated eigendirections allows us to detect changes in dynamical behavior and the underlying intrinsic system dimensionality directly from data.