  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 node attributes. We apply our framework
to graph classification problems and obtain performances competitive with other
persistence-based architectures. To provide the underlying theoretical
foundations, we extend the differentiability result for ordinary persistent
homology to extended persistent homology.
