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.
翻译:下游抽样生成了粗化的、多分辨率的数据表达方式,这些数据被用于,例如,对大型图像进行失落压缩和直观化,降低计算成本,促进深神经代表性学习。不幸的是,由于缺乏常规结构,对于下游抽样应如何应用于图表和链接数据仍没有共识。事实上,上述目标仍然需要减少图形数据,但减少机制对维护地形结构和特性没有同样的关注,同时允许分辨率调整,如常规数据下游。在本文中,我们朝这个方向迈出了一步,在常规和图表数据中引入了对下游抽样的统一解释。特别是,我们定义了一个图表粗略分析机制,它是固定数据中可控缩微缩缩缩机制的图形结构对应机制。我们证明,在路径长度上存在扭曲的理论保证,以及保留关键地形特性的能力,正如常规数据下游数据所做的那样。我们利用这些概念来定义一个图表集合机制,我们在图表分类中从经验上评估了对下层抽样抽样数据进行解释,提供了一种平行的算法,从而使得在图表分类中能够进行高效的聚合性分析。