Apparaît dans la collection : A Multiscale tour of Harmonic Analysis and Machine Learning - To Celebrate Stéphane Mallat's 60th birthday

I will discuss multiscale basis dictionaries, in particular, the Hierarchical Graph Laplacian Eigen Transform (HGLET) and the Generalized Haar-Walsh Transform (GHWT), which my group originally developed for analyzing signals measured on nodes of an input graph, but which we have recently generalized for signals defined on edges, triangles, tetrahedra, etc. of a given simplicial complex using the Hodge Laplacians. These dictionaries consist of redundant sets of multiscale basis vectors and the corresponding expansion coefficients of a given signal. These provide a large number of orthonormal bases among which one can select the most suitable basis for one's task using the best-basis algorithm and its relatives. I will also discuss how to construct scattering networks for signals on simplicial complexes using the HGLET and the GHWT. Our new scattering networks cascade the moments (up to fourth order) of the modulus of the dictionary coefficients followed by the local averaging process. Consequently, the resulting features are robust to perturbations of input signals and invariant w.r.t. node permutations. I will demonstrate the usefulness of these dictionaries using the coauthorship/citation complex and the Science News article classification. This is joint work with Stefan Schonsheck and Eugene Shvarts.