Finding Principal Paths in Data Space

In this paper, we introduce the concept of principal paths in data space; we show that this is a well-characterized problem from the point of view of cognition, and that it can lead to salient insights in the analyzed data enabling topological/holistic descriptions. These paths, interestingly, can be interpreted as local principal curves, and in this paper, we suggest that they are analogous to what, in the statistical mechanics realm, are called minimum free-energy paths. Here, we move that concept from physics to data space and compute them in both the original and the kernel space. The algorithm is a regularized version of the well-known k-means clustering algorithm. The regularization parameter is derived via an in-sample model selection process based on the Bayesian evidence maximization. Interestingly, we show that this choice for the regularization parameter consistently leads to the same manifold even when changing the number of clusters. We apply the method to common data sets, dynamical systems, and, in particular, to molecular dynamics trajectories showing the generality, the usefulness of the approach and its superiority with respect to other related approaches.