  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.
