In graph learning, maps between graphs and their subgraphs frequently arise. For instance, when coarsening or rewiring operations are present along the pipeline, one needs to keep track of the corresponding nodes between the original and modified graphs. Classically, these maps are represented as binary node-to-node correspondence matrices and used as-is to transfer node-wise features between the graphs. In this paper, we argue that simply changing this map representation can bring notable benefits to graph learning tasks. Drawing inspiration from recent progress in geometry processing, we introduce a spectral representation for maps that is easy to integrate into existing graph learning models. This spectral representation is a compact and straightforward plug-in replacement and is robust to topological changes of the graphs. Remarkably, the representation exhibits structural properties that make it interpretable, drawing an analogy with recent results on smooth manifolds. We demonstrate the benefits of incorporating spectral maps in graph learning pipelines, addressing scenarios where a node-to-node map is not well defined, or in the absence of exact isomorphism. Our approach bears practical benefits in knowledge distillation and hierarchical learning, where we show comparable or improved performance at a fraction of the computational cost.

翻译：在图学习过程中，原始图与子图之间的映射关系频繁出现。例如，当处理流程中包含粗化或重连操作时，需要追踪原始图与修改后图之间的对应节点。传统上，这些映射被表示为二值化节点对应矩阵，并直接用于跨图传递节点特征。本文论证改变这种映射表示方式能为图学习任务带来显著益处。受几何处理领域最新进展启发，我们提出一种易于集成到现有图学习模型的谱映射表示方法。该谱表示形式简洁紧凑，可直接作为即插即用组件，且对图的拓扑变化具有稳健性。值得注意的是，该表示展现出具有可解释性的结构特性，与光滑流形上的近期研究成果形成类比。我们通过实验验证了在图学习流程中引入谱映射的优势，特别是处理节点间映射不明确或缺乏精确同构的场景。该方法在知识蒸馏和层次学习场景中具有实用价值，能以较低计算成本实现相当或更优的性能表现。