Geometric Data Analysis of Point Clouds
A theme of topological data analysis is the construction of a graph whose nodes are data points and edges represent affinity between nodes. Persistent homology, for example, computes homology of the Vietoris-Rips complex of a series of graphs as a scale parameter is varied. In this talk, we take a more geometric approach by assuming the data points lie on a Riemannian manifold, and reconstruct the Laplace-Beltrami operator. In particular, we show how build a graph Laplacian that converges, pointwise and spectrally, to the continuous operator in the large data limit. If this can be achieved, topological information about the manifold is contained in a single graph, unlike the persistent homology approach. Since real data is typically sampled irregularly, it is necessary to introduce a criterion called Continuous k-Nearest Neighbors (CkNN) for the graph construction that implies convergence for arbitrary sampling.
Refreshments in Math Commons Room at 3:30pm