Graph coarsening aims to diminish the size of a graph to lighten its memory footprint, and has numerous applications in graph signal processing and machine learning. It is usually defined using a reduction matrix and a lifting matrix, which, respectively, allows to project a graph signal from the original graph to the coarsened one and back. This results in a loss of information measured by the so-called Restricted Spectral Approximation (RSA). Most coarsening frameworks impose a fixed relationship between the reduction and lifting matrices, generally as pseudo-inverses of each other, and seek to define a coarsening that minimizes the RSA. In this paper, we remark that the roles of these two matrices are not entirely symmetric: indeed, putting constraints on the lifting matrix alone ensures the existence of important objects such as the coarsened graph's adjacency matrix or Laplacian. In light of this, in this paper, we introduce a more general notion of reduction matrix, that is not necessarily the pseudo-inverse of the lifting matrix. We establish a taxonomy of ``admissible'' families of reduction matrices, discuss the different properties that they must satisfy and whether they admit a closed-form description or not. We show that, for a fixed coarsening represented by a fixed lifting matrix, the RSA can be further reduced simply by modifying the reduction matrix. We explore different examples, including some based on a constrained optimization process of the RSA. Since this criterion has also been linked to the performance of Graph Neural Networks, we also illustrate the impact of this choices on different node classification tasks on coarsened graphs.

翻译：图粗化旨在减小图的规模以降低其内存占用，在图信号处理与机器学习领域具有广泛应用。该方法通常通过约简矩阵与提升矩阵定义：前者将图信号从原图投影至粗化图，后者则实现反向投影。这一过程会导致信息损失，其度量标准即所谓的受限谱逼近。多数粗化框架强制约简矩阵与提升矩阵之间存在固定关系（通常互为伪逆），并寻求最小化受限谱逼近的粗化方案。本文指出，这两类矩阵的作用并非完全对称：事实上，仅对提升矩阵施加约束即可确保粗化图的邻接矩阵或拉普拉斯矩阵等重要对象的存在性。鉴于此，本文引入更广义的约简矩阵概念，其不必是提升矩阵的伪逆。我们建立了“可容许”约简矩阵族的分类体系，探讨了各类矩阵需满足的性质及其闭式描述的存在性。研究表明，对于由固定提升矩阵表示的特定粗化方案，仅通过修改约简矩阵即可进一步降低受限谱逼近。我们探究了若干实例，包括基于受限谱逼近约束优化过程的案例。由于该准则亦与图神经网络的性能相关联，本文还通过粗化图上的节点分类任务，阐释了约简矩阵选择对模型性能的影响。