Optimisation of Spectral Wavelets for Persistence-based Graph Classification

Abstract

A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimises the choice of wavelet for a dataset of graphs, such that their associated persistence diagrams capture features of the graphs that are best suited to a given data science problem. Since the spectral wavelet signature of a graph is derived from its Laplacian, our framework encodes geometric properties of graphs in their associated persistence diagrams and can be applied to graphs without a priori vertex features. We demonstrate how our framework can be coupled with different persistence diagram vectorisation methods for various supervised and unsupervised learning problems, such as graph classification and finding persistence maximising filtrations, respectively. To provide the underlying theoretical foundations, we extend the differentiability result for ordinary persistent homology to extended persistent homology.Recently, persistent homology [ZC05; EH08] has been applied as a feature map that explicitly represents topological and geometric features of a graph as a set of persistence diagrams (a.k.a. barcodes). In the context of our discussion, the persistent homology of a graph G = (V, E) depends on a vertex function f : V → R. In the case where a vertex function is not given with the data, several schemes have been proposed in the literature to assign vertex functions to graphs in a consistent way. For example, vertex functions can be constructed using local geometric descriptions of vertex neighbourhoods, such as discrete curvature [ZW19] and heat kernel signatures [Car+20b].