TwinS-GCN：谱图卷积网络的谱共轭
TwinS-GCN: Spectral conjugate for Spectral Graph Convolutional Networks
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中文总结 AI 辅助
针对图卷积网络在长程依赖与过平滑间的矛盾，提出基于谱共轭的TwinS-GCN，通过非全纯核构建求和与比率滤波器，实现低深度下高效图迁移与节点分类。
中文摘要 AI 辅助
图卷积网络通过图移位算子进行重复的局部聚合来传播信息；即，一个$K$层网络可到达$K$跳邻域。一方面，这种传播可能导致过平滑。另一方面，长程依赖需要网络深度。在长距离上无衰减地传输信息要求移位算子能够区分流动方向，对称算子无法做到这一点，但有向算子可以。提取纯方向性的一种自然方式是通过笛卡尔分解取移位算子的反对称部分，然而，该部分通常不与移位算子本身交换，这意味着基于它构建的滤波器不具有移位不变性。我们转而使用谱共轭，即算子在映射$\tau:z\mapsto \bar{z}$下的像，它与移位算子交换，并将其分解为耗散部分和非耗散部分。由此产生两类滤波器：求和滤波器，其非耗散分量在无能量损失的情况下传输信号；以及比率滤波器，即分量对之间的比率而非移位算子的多项式。两者均源于非全纯核，使其位于经典谱卷积所基于的全纯类之外。在有向环上，比率滤波器成为具有全局脉冲响应的IIR滤波器，我们证明了它与每个$K$次多项式滤波器相比存在长程可达差距。切比雪夫重参数化提供了具有实系数的稳定顶点域层，从而得到TwinS-GCN，它能在减少深度的情况下解决图迁移任务，并在节点分类基准上与最先进的图卷积网络竞争。
英文摘要
Graph convolutional networks propagate information by repeated local aggregation through a graph shift operator; i.e., a $K$-layer network reaches $K$ hops neighborhood. On the one hand, such spreading can lead to oversmoothing. On the other hand, long-range dependencies demand the depth. Transporting information on long distances and without attenuation requires the shift to distinguish a direction of flow, which a symmetric operator cannot perform but a directed one can fulfill. A natural way to extract pure-directionality is to take the skew-symmetric part of the shift operator through the Cartesian split, which, however, generally does not commute with the shift itself, meaning that the filters built on it are not shift-invariant. We instead use the spectral conjugate; i.e., the image of the operator under $τ:z\mapsto \bar{z}$, which commutes with the shift and splits it into a dissipative and a non-dissipative part. Two filter families follow: a sum filter, whose non-dissipative component transports signal without energy loss, and a ratio filter, ratio in the pair of components rather than polynomial in the shift. Both arise from non-holomorphic kernels, placing them outside the holomorphic class underlying classical spectral convolution. On the directed cycle, the ratio filter becomes an IIR filter with global impulse response, for which we prove a long-range reach gap against every degree-$K$ polynomial filter. Chebyshev reparameterization gives stable vertex-domain layers with real coefficients, yielding TwinS-GCN, which solves graph transfer tasks at reduced depth and is competitive with state-of-the-art graph convolutional networks on node classification benchmarks.
