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.

翻译：图形的光谱波子签名决定了一个过滤器, 并因此决定了一组相关的延长的持久性图表。 我们提出了一个框架, 选择对图表数据集的波子选择, 以便它们的相关持久性图表能够捕捉最适合特定数据科学问题的图形特征。 由于图的光谱波子签名取自于它的 Laplacian, 我们的框架将相关持久性图表的几何特性编码起来, 并且可以应用到没有前置节点属性的图形中。 我们应用了我们的框架来图形分类问题, 并获得与其他基于持久性的建筑的性能竞争力。 为了提供理论基础, 我们将普通持久性同系的可变性结果扩展到扩展的持久性同系。