Density ridge manifold traversal

The density ridge framework for estimating principal curves and surfaces has in a number of recent works been shown to capture manifold structure in data in an intuitive and effective manner. However, to date there exists no efficient way to traverse these manifolds as defined by density ridges. This is unfortunate, as manifold traversal is an important problem for example for shape estimation in medical imaging, or in general for being able to characterize and understand state transitions or local variability over the data manifold. In this paper, we remedy this situation by introducing a novel manifold traversal algorithm based on geodesics within the density ridge approach. The traversal is executed in a subspace capturing the intrinsic dimensionality of the data using dimensionality reduction techniques such as principal component analysis or kernel entropy component analysis. A mapping back to the ambient space is obtained by training a neural network. We compare against maximum mean discrepancy traversal, a recent approach, and obtain promising results.