Finding the Homology of Manifolds using Ellipsoids

Abstract

A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C 1 -submanifold without boundary in a Euclidean space. We show that we can deformation retract the union of ellipsoids, centered at sample points and stretching in the tangent directions, to the manifold. Hence the homotopy type, and therefore also the homology type, of the manifold is the same as that of the nerve complex of the cover by ellipsoids. By thickening sample points to ellipsoids rather than balls, our results require a smaller sample density than comparable results in the literature. They also advocate using elongated shapes in the construction of barcodes in persistent homology.

A Family of Density-Scaled Filtered Complexes

“…One idea for improving on the current definition is to consider the local covariance of the probability distribution at each point [35] and to modify the Riemannian metric in such a way that with respect to the new Riemannian metric, the local covariance matrix at each point is the identity matrix (with respect to a positively oriented orthonormal basis). This idea is akin to the usual normalization that data scientists often do in Euclidean space, and it is also reminiscent of the ellipsoidthickenings of [26]. In the k-nearest neighbors filtration, each point is connected to its k-nearest neighbors at the kth filtration step.…”