HAQJSK: Hierarchical-Aligned Quantum Jensen-Shannon Kernels for Graph Classification

In this work, we propose two novel quantum walk kernels, namely the Hierarchical Aligned Quantum Jensen-Shannon Kernels (HAQJSK), between un-attributed graph structures. Different from most classical graph kernels, the proposed HAQJSK kernels can incorporate hierarchical aligned structure information between graphs and transform graphs of random sizes into fixed-size aligned graph structures, i.e., the Hierarchical Transitive Aligned Adjacency Matrix of vertices and the Hierarchical Transitive Aligned Density Matrix of the Continuous-Time Quantum Walks (CTQW). With pairwise graphs to hand, the resulting HAQJSK kernels are defined by computing the Quantum Jensen-Shannon Divergence (QJSD) between their transitive aligned graph structures. We show that the proposed HAQJSK kernels not only reflect richer intrinsic whole graph characteristics in terms of the CTQW, but also address the drawback of neglecting structural correspondence information that arises in most R-convolution graph kernels. Moreover, unlike the previous QJSD based graph kernels associated with the QJSD and the CTQW, the proposed HAQJSK kernels can simultaneously guarantee the properties of permutation invariant and positive definiteness, explaining the theoretical advantages of the HAQJSK kernels. The experiment indicates the effectiveness of the new proposed kernels.