Joshua Cape

Networks, matrices, and geometry: the generalized random dot product graph model

We introduce the generalized random dot product graph model, a latent space network model that provides a unified setting in which to study spectral clustering methods, factorizations of matrices, and the geometry of point cloud configurations. We establish that, for both the normalised Laplacian and adjacency matrix, the vector representations of nodes obtained by spectral embedding provide strongly consistent latent position estimates with asymptotically Gaussian error. Direct methodological consequences follow from the observation that the mixed membership and standard stochastic block models are special cases where the latent positions live respectively inside or on the vertices of a simplex. Hence, estimation via spectral embedding can be achieved by estimating the simplicial support or by fitting a Gaussian mixture model, respectively. We highlight applications of our results to the analysis of cybersecurity data and connectomics.
Furthermore, we formalize that spectral embedding estimates, which are computationally tractable, are also statistically admissible for stochastic block model graphs, even when compared to purportedly optimal, computationally intractable maximum likelihood estimation under no rank assumption. Our findings resolve a gap and potential misunderstanding in the established literature.
This talk is based on joint work with Patrick Rubin-Delanchy, Minh Tang, and Carey E. Priebe.

Joshua is a NSF Postdoctoral Research Fellow from the University of Michigan, with the top Annals of Stats paper https://arxiv.org/abs/1705.10735. Joshua will be giving a Statistical Seminar during his visit with us at the Heilbronn Institute on Thursday 19th December. To read more on Joshua Cape please see his webpage.