Machine learning for node classification on graphs is a prominent area driven by applications such as recommendation systems. State-of-the-art models often use multiple graph convolutions on the data, as empirical evidence suggests they can enhance performance. However, it has been shown empirically and theoretically, that too many graph convolutions can degrade performance significantly, a phenomenon known as oversmoothing. In this paper, we provide a rigorous theoretical analysis, based on the two-class contextual stochastic block model (CSBM), of the performance of vanilla graph convolution from which we remove the principal eigenvector to avoid oversmoothing. We perform a spectral analysis for $k$ rounds of corrected graph convolutions, and we provide results for partial and exact classification. For partial classification, we show that each round of convolution can reduce the misclassification error exponentially up to a saturation level, after which performance does not worsen. We also extend this analysis to the multi-class setting with features distributed according to a Gaussian mixture model. For exact classification, we show that the separability threshold can be improved exponentially up to $O({\log{n}}/{\log\log{n}})$ corrected convolutions.

翻译：图节点分类的机器学习是一个由推荐系统等应用驱动的重要领域。最先进的模型通常对数据使用多次图卷积，因为经验证据表明这能提升性能。然而，已有实证和理论研究表明，过多的图卷积会显著降低性能，这种现象称为过平滑。本文基于二类上下文随机块模型（CSBM），对去除主特征向量以避免过平滑的原始图卷积性能进行了严格的理论分析。我们对$k$轮修正图卷积进行了谱分析，并给出了部分分类和精确分类的结果。对于部分分类，我们证明每轮卷积能以指数方式降低误分类误差直至饱和水平，此后性能不再恶化。我们还将此分析扩展至特征服从高斯混合模型的多类场景。对于精确分类，我们证明可分离性阈值可通过最多$O({\log{n}}/{\log\log{n}})$轮修正卷积获得指数级改善。