Generalizing Downsampling from Regular Data to Graphs

Downsampling produces coarsened, multi-resolution representations of data and it is used, for example, to produce lossy compression and visualization of large images, reduce computational costs, and boost deep neural representation learning. Unfortunately, due to their lack of a regular structure, there is still no consensus on how downsampling should apply to graphs and linked data. Indeed reductions in graph data are still needed for the goals described above, but reduction mechanisms do not have the same focus on preserving topological structures and properties, while allowing for resolution-tuning, as is the case in regular data downsampling. In this paper, we take a step in this direction, introducing a unifying interpretation of downsampling in regular and graph data. In particular, we define a graph coarsening mechanism which is a graph-structured counterpart of controllable equispaced coarsening mechanisms in regular data. We prove theoretical guarantees for distortion bounds on path lengths, as well as the ability to preserve key topological properties in the coarsened graphs. We leverage these concepts to define a graph pooling mechanism that we empirically assess in graph classification tasks, providing a greedy algorithm that allows efficient parallel implementation on GPUs, and showing that it compares favorably against pooling methods in literature. * We would like to thank Federico Poloni and Federico Errica for their most useful suggestions on earlier versions of this paper.