Spectral Methods of Graphs for Data Sets of Mixed Dimensions

High dimensional data often consist of parts with different intrinsic dimension. We study how spectral methods on graphs adapt to data containing intersecting pieces of different dimensions. We show that unnormalized Laplacian on random geometric (and similar) graphs only sees the variations in the highest dimension, while appropriately normalized Laplacian converges to Laplace-Beltrami operator in all dimensions simultaneously. Somewhat surprisingly, we show that graph Laplacians based on kNN graphs only see the variations in the lowest dimension. We thus suggest new normalizations that allow for variations in all dimensions to be captured. For intersecting manifolds we identify when and how is the information transferred between manifolds. The talk is based on joint work with Leon Bungert and Simone Di Marino.