PoincareNorm: Rethinking Over-smoothing beyond Dirichlet energy

Weiqi Guan, Zihao Shi

Keywords: over-smoothing, node similarity measure，normalization。

TL;DR: we propose a series of node similarity measures which generize Dirichlet energy and a normalization called PoincareNorm which generizes PairNorm.

Abstract:

Dirichlet energy is intuitive and commonly used to measure over-smoothing. However, Dirichlet energy can only capture information about the first-order derivative of features. In light of this, we propose a series of node similarity measures which are the energy of higher-order derivatives of features and generalize Dirichlet energy. After we rigorously analyze the property of proposed measures and its application to establish the sharp decay rate of Dirichlet energy under continuous diffusion or discrete random walk which is closely related to the first nonzero eigenvalue of graph Laplacian. Lastly, to address over-smoothing with respect to these measures, we propose a normalization termed PoincareNorm which generalizes PairNorm to control our proposed measures. We consider the semi-supervised node classification task in the scenario without missing features, PoincareNorm outperforms existing normalization methods.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 5273