Scattering Networks on Simplicial Complexes Using Multiscale Basis Dictionaries

Naoki Saito, Stefan C Schonsheck, Eugene Shvarts

Abstract:

We discuss our new scattering networks for signals on simplicial complexes. Our construction is based on multiscale basis dictionaries on simplicial complexes, i.e., the $\kappa$-GHWT and $\kappa$-HGLET, which we recently developed for simplices of dimension $\kappa \in \mathbb{N}$ in a given simplicial complex by generalizing the node-based Generalized Haar-Walsh Transform (GHWT) and Hierarchical Graph Laplacian Eigen Transform (HGLET). The $\kappa$-GHWT and the $\kappa$-HGLET both form redundant sets (i.e., dictionaries) of multiscale basis vectors and the corresponding expansion coefficients of a given signal. Then, our new scattering network cascades the moments (up to fourth order) of the modulus of the dictionary coefficients within the basis dictionaries 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.e., they are effective for classifying signals recorded on $\kappa$-simplices of a given simplicial complex.

We will demonstrate such effectiveness in document type classification using the Science News database. More precisely, using the word2vec method, we generate the nodes corresponding to a selected set of words
(terms) in this database and form a $k$-nearest neighbor graph, which is viewed
as a simplicial complex.
Such a graph clearly contains $\kappa$-simplices, e.g., triangles ($\kappa=2$),
tetrahedra ($\kappa=3$), etc., which correspond to combinations of $\kappa+1$
words. Each document leads to a signal for $\kappa$-simplices of this graph.
That is, the value of a node is the frequency of occurrence of the word
corresponding to that node ($\kappa=0$) while the value of a $\kappa$-simplex
is the sum of the values of the $\kappa+1$ nodes that form that $\kappa$-simplex.
In this way, a signal on $\kappa$-simplices reflects the frequency of mutual occurrences of $\kappa+1$ words, which may better characterize the nature of the corresponding document.

We also discuss the similarities and differences between our proposed method and the previously-proposed methods such as the Deep Haar Scattering Networks of Cheng et al., Geometric Scattering Networks of Gao et al., and Hodgelets of Roddenberry et al.

Submission Type: Abstract