Harnessing spectral representations for subgraph alignment

Marco Pegoraro, Riccardo Marin, Arianna Rampini, Simone Melzi, Luca Cosmo, Emanuele Rodolà

Keywords: Graph alignment, Spectral theory

Abstract: With the rise and advent of graph learning techniques, graph data has become ubiquitous in the machine learning field. However, while several efforts have been devoted to the design of new convolutional architectures, pooling or positional encoding schemes, relatively little focus has been spent on modeling pairwise problems such as signal transfer, graph isomorphism and subgraph correspondence tasks. With this paper, we anticipate the need for a convenient framework to deal with problems that revolve around the notion of a map among graphs, and focus in particular on the challenging subgraph alignment scenario. We claim that, first and foremost, the representation of a map plays a central role in how these problems should be modeled -- be it a map inference problem or a simpler signal transport task. Taking the hint from recent work in geometry processing, we propose the adoption of a spectral representation for maps that is compact, easy to compute, permutation-equivariant, easy to plug into learning pipelines, and especially effective for a wide range of situations, most notably when dealing with subgraph alignment problems. We further report for the first time a surprising phenomenon where the partiality arising in subgraph alignment is manifested in the structure of the map coefficients, even in the absence of exact isomorphism, and which is consistently observed over different families of graphs.

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