The study of high dimensional data sets often rely on their low dimensional projections that preserve the local geometry of the original space. While numerous methods have been developed to summarize this space as variations of tree-like structures, they are usually non-parametric and "static" in nature. As data may come from systems that are dynamical such as a differentiating cell, a static, non-parametric characterization of the space may not be the most appropriate. Here, we developed a framework, the principal flow, that is capable of characterizing the space in a dynamical manner. The principal flow, defined using neural ODEs, directs motion of a particle through the space, where the trajectory of the particle resembles the principal curve of the dataset. We illustrate that our framework can be used to characterize shapes of various complexities, and is flexible to incorporate summaries of relaxation dynamics.
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