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Whilst spectral Graph Neural Networks (GNNs) are theoretically well-foundedin the spectral domain, their practical reliance on polynomial approximationimplies a profound linkage to the spatial domain. As previous studies rarelyexamine spectral GNNs from the spatial perspective, their spatial-domaininterpretability remains elusive, e.g., what information is essentially encodedby spectral GNNs in the spatial domain? In this paper, to answer this question,we establish a theoretical connection between spectral filtering and spatialaggregation, unveiling an intrinsic interaction that spectral filteringimplicitly leads the original graph to an adapted new graph, explicitlycomputed for spatial aggregation. Both theoretical and empirical investigationsreveal that the adapted new graph not only exhibits non-locality but alsoaccommodates signed edge weights to reflect label consistency between nodes.These findings thus highlight the interpretable role of spectral GNNs in thespatial domain and inspire us to rethink graph spectral filters beyond thefixed-order polynomials, which neglect global information. Built upon thetheoretical findings, we revisit the state-of-the-art spectral GNNs and proposea novel Spatially Adaptive Filtering (SAF) framework, which leverages theadapted new graph by spectral filtering for an auxiliary non-local aggregation.Notably, our proposed SAF comprehensively models both node similarity anddissimilarity from a global perspective, therefore alleviating persistentdeficiencies of GNNs related to long-range dependencies and graph heterophily.Extensive experiments over 13 node classification benchmarks demonstrate thesuperiority of our proposed framework to the state-of-the-art models.

Rethinking Spectral Graph Neural Networks with Spatially Adaptive Filtering