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A graph's spectral wavelet signature determines a filtration, and consequently an associated set of extended persistence diagrams. We propose a framework that optimizes 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.

Recently, two approaches were suggested which combine signed particles and neural networks to speed up the time-dependent simulation of quantum systems. Both specialize on the efficient computation of a multi-dimensional function defined over the phase space known as the Wigner kernel. While the first approach completely defines the network analytically, the second is based on an architecture with generalization capabilities. Although relatively simple, these networks can reduce the amount of memory needed and provide a computational speedup, but they are both affected by the use of expensive activation functions (sinusoidals). In this work, we go beyond these previous strategies and suggest a more general network consisting of a set of different hidden layers which are based on less expensive activation functions (e.g. rectified linear units), and which now predict one column of the discretized kernel at a time. This approach comes with generalization capabilities and allows the network to accurately learn a transform from the space of potentials to the space of kernels. As it is shown in our final validation test, this new approach performs very well during the simulation of quantum systems. In fact, while keeping a good accuracy, it further reduces the amount of memory required along with its computational burden.