Speaker. Naoki Saito, Department of Mathematics, UC-Davis.

Title. Multiscale Hodge Scattering Networks for Data Analysis 

Abstract. I will discuss Multiscale Hodge Scattering Networks (MHSNs), a new method to extract features from signals measured on simplicial complexes, which we recently developed. Our construction is based on multiscale basis dictionaries on simplicial complexes, i.e., the k-GHWT and k-HGLET, which we developed for simplices of dimension k in a given simplicial complex by generalizing the node-based Generalized Haar-Walsh Transform (GHWT) and Hierarchical Graph Laplacian Eigen Transform (HGLET). The k-GHWT and the k-HGLET both form redundant sets (i.e., dictionaries) of multiscale basis vectors and the corresponding expansion coefficients of a given signal. Our MHSNs use a layered structure analogous to a convolutional neural network (CNN) to cascade the moments of the modulus of the dictionary coefficients. The resulting features are invariant to reordering of the simplices (i.e., node permutation of the underlying graphs). Importantly, the use of multiscale basis dictionaries in our MHSNs admits a natural pooling operation that is akin to local pooling in CNNs, and which may be performed either locally or per-scale. As a result, we are able to extract a rich set of descriptive yet robust features that can be used along with very simple machine learning methods (i.e., logistic regression or support vector machines) to achieve high-accuracy classification systems with far fewer number of parameters to train than most modern graph neural networks. Finally, we demonstrate the usefulness of our MHSNs in three distinct types of problems: signal classification, domain (i.e., graph/simplex) classification, and molecular dynamics prediction.This is joint work with Drs. Stefan Schonsheck and Eugene Shvarts.