Scientific Computation and Applied & Industrial Mathematics: David Gleich

Higher-order methods that use multiway and multilinear correlations are necessary to identify important structures in complex data from biology, neuroscience, ecology, systems engineering, and sociology. We will study our recent generalization of spectral clustering to higher-order structures in depth. This will include a generalization of the Cheeger inequality (a concise statement about the approximation quality) to higher-order structures in networks including network motifs. This is easy to implement and seamlessly enables spectral clustering-style methods for directed, signed, and many other types of complex networks. If there is time, we will see software demonstrations in the Julia language for reproducibility. I will also briefly highlight recent methods we have developed that use new types of stochastic processes and random walks to study these data involving tensor eigenvectors. These topics motivate a number of exciting open questions at the intersection of numerical linear algebra, optimization, and data analysis.