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Nonlinear manifold learning algorithms, such as diffusion maps, have beenfruitfully applied in recent years to the analysis of large and complex datasets. However, such algorithms still encounter challenges when faced with realdata. One such challenge is the existence of "repeated eigendirections," whichobscures the detection of the true dimensionality of the underlying manifoldand arises when several embedding coordinates parametrize the same direction inthe intrinsic geometry of the data set. We propose an algorithm, based on locallinear regression, to automatically detect coordinates corresponding torepeated eigendirections. We construct a more parsimonious embedding using onlythe eigenvectors corresponding to unique eigendirections, and we show that thisreduced diffusion maps embedding induces a metric which is equivalent to thestandard diffusion distance. We first demonstrate the utility and flexibilityof our approach on synthetic data sets. We then apply our algorithm to datacollected from a stochastic model of cellular chemotaxis, where our approachfor factoring out repeated eigendirections allows us to detect changes indynamical behavior and the underlying intrinsic system dimensionality directlyfrom data.

Parsimonious Representation of Nonlinear Dynamical Systems Through Manifold Learning: A Chemotaxis Case Study