The Multi-Orbits Skew Spectrum: Boosting Permutation-Invariant Data Representations

Armando Bellante, Martin Plávala, Alessandro Luongo

Keywords: group theory, representation theory, Fourier, feature mapping, permutation invariant, graph kernels, skew spectrum

TL;DR: A group-theoretical permutation-invariant feature mapping for labeled graphs, multigraphs, and hypergraphs.

Abstract: We generalize the concept of skew spectrum of a graph, a group-theoretical permutation-invariant feature mapping. The skew spectrum considers adjacency matrices as functions over $\mathbb{S}_n$ and leverages Fourier transform and group-theoretical tools to extract features that are invariant under the group action. The main shortcoming of the previous result is that the skew spectrum only works for unlabeled graphs. The reason is that these graphs can be represented using matrices whose main diagonal contains zeros, meaning that there is only one set of elements that can permute among themselves (i.e., one orbit). However, the representations of more complex graphs (e.g., labeled graphs, multigraphs, or hypergraphs) have different sets of elements that can consistently permute on different orbits. In this work, we generalize the skew spectrum to the multiple orbits case. Our multi-orbits skew spectrum produces features invariant to such permutations and possibly informative of non-consistent ones. We show this method can improve the performances of models that learn on graphs by providing comparisons with single orbit representations and eigenvalues. Moreover, the theory is general enough to handle invariance under the action of any finite group on multiple orbits and has applications beyond the graph domain.

Type Of Submission: Extended abstract (max 4 main pages).

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Type Of Submission: Extended abstract.