maps of Hessian eigenvalues for gap-finding

As my loyal reader knows, Gaby Contardo (Flatiron) and I have been looking for gaps (valleys, voids) in point clouds using geometric methods on density estimates. Today she just did the very simplest thing of estimating the largest eigenvalue of the second-derivative tensor (Hessian of density with respect to position), and visualizing it for different density estimates (different bandwidths) and different bootstrap resamplings of the data. It is obvious, looking at these plots, that we can combine these maps into good gap-finders! This is simpler than our previous approaches, and will generalize better to higher dimensions. It's also slow, but we don't see anything that can be fast, especially in “high” dimensions (high like 3 or 4!!).