Title: Taxonomy of reduction matrices for Graph Coarsening
Venue: NeurIPS 2025
Accepted: yes
Keywords: graph coarsening

graph machine learning
Abstract: 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nGraph coarsening is an important algorithmic primitive for many pipelines involving graphs in ML, such as graph/geometric deep learning, but also computational methods working on or with graphs beyond ML. There are celebrated results on spectral approximations of large graphs by sparser ones (see the work of Spielman for instance) but fewer strong theoretical results on reducing/coarsening large graphs to smaller ones with mathematical guarantees. \n\nIn ML, it is of particular importance to preserve features defined over the graph and that can represent external signals or graph structure. This paper builds on and extends previous work by Lukas, widening the coarsening and lifting operations that lead to preserved spectral properties (defined via a stable embedding condition). A systematic exploration of the \"design space\" of these operations allow the authors to propose several new operations and recover older ones. \n\nExperimental validation suggest that these techniques are particularly suited for homophilic/smooth signals, but less so when this restriction is lifted.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- very clearly written, the paper is easy to follow with crystal motivations\n- strong mathematical results, even if limited by the spectral objectives (which one must say are the norm in this field)\n\nWeaknesses:\n- how important is the structure of the graph ?\n- it is hard to tell whether the theoretical motivation results in practical gains \n- practical results are limited to rather small graphs where one could question the usefulness of the whole endeavour\n\nQUESTIONS:\n- Please clarify the role of graph structure (beyond the example of the random geometric graph) and graph signal models (is the method suited for non-smooth graph signals at all)\n- Given the importance of Kron reduction in many fields (it has been used in graph signal processing/ML), and its well-studied spectral properties, I think it deserves a more elaborate discussion: can you compare both approaches, what would be the advantages/disadvantages of both. Kron reduction can also be (at least partially) inverted using its spectral approximation properties, is there any link to what is being proposed in this paper ?\n\nLIMITATIONS:\nYes, except maybe on the practical limitations of the methods. I do not consider this critical though.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the admission space of reduction matrices and its optimization towards preserving Restricted Spectral Approximation (RSA) in graph coarsening. Specifically, the authors observe that the coarsening can be solely defined by the lifting matrix and then decouple the relation between the lifting and reduction matrix, which is typically chosen as pseudo-inverse of each other. With this, they reveal three admission spaces of the reduction matrix, from projection to generalized reflexive inverse and to generalized reflexive inverse of the same support. They then study optimizing the reduction matrix for the RSA objective and discuss six reduction matrices under different admission spaces and sparsity constraints. Experiments have been conducted on two small datasets to demonstrate the RSA preservation for different reduction matrices and their impacts on the accuracy of the GNN application.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\nThe paper establishes a good understanding between the relations between the lifting and reduction matrices and pointed out the lifting matrix defines the coarsening while the reduction matrix can affect the RSA preservation and thus can be optimized towards a better preservation. It is a solid paper that builds on top of the RSA objective and can be of great reference for similar research in graph coarsening.\n\nAnalyze the three levels of admission spaces for the reduction matrix and formulate the optimization over the admission spaces. Discuss the optimization and special properties of each reduction matrix, ranging from typical Moore-Penrose inverse to RSA optimization with a sparsity constraint. Although I did not review the proof in Appendix, the conclusions look natural and should be correct.\n\nWhile the main contribution is in theory, the authors have performed experiments to study the empirical performance on the RSA optimization and the impacts on downstream GNN application.\n\nWeaknesses:\n\nThe spectral guarantee using RSA proposed in 2019 is only one type of key properties graph coarsening approximates. It is not clear how the developed optimization and coarsening framework have significant impacts on other types of graph coarsening objective.\n\nThe experiments were conducted on small-scale graphs, while this is a common problem for spectral methods. The empirical computation costs for different optimizations are missing and should be reported.\n\nQUESTIONS:\nComments:\n\nThe lifting and reduction matrices are often considered to be pseudo-inverse of each other with the intuition that the two processes are inverse. This paper defines the adjacency matrix of the coarsened graph as (1) and optimizes P. Would it be possible to use P to define the coarsened graph and then optimize Q for the RSA target instead? \n\nThe well-partitioned property may not be always true in real-life coarsening tasks with possibly overlapping clusters/supernodes. What are known results for the overlapping setting?\n\nThe computation overhead for getting different reduction matrices should be provided to offer guidance in the selection.\n\nI’m quite curious on the empirical performance of GNN on the RSA approximation and data-driven graph pooling methods [39,13,34]. RSA may have better generalization on different downstream tasks while data-drive methods are optimized for specific task and data.\n\nTypo: Line 135: a eigen-subspace.\n\nLine 43: “In virtually all works on graph coarsening, these two matrices are simply pseudo-inverse of each other, and both can represent the “graph coarsening” indifferently.” and Line 85: “These methods use interchangeably the reduction and the lifting matrix by choosing one as the transposed of the other.” Graph coarsening and pooling seem to be similar, if not the same. Why does one set the two matrices to be pseudo-inverse of each other and the other set them to be transposed of each other? Please clarify.\n\nLIMITATIONS:\nI did not see the discussion but this work may not have clear negative societal impacts that have to be explained. Should add limitation discussions though.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a taxonomy of admissible families of reduction matrices for graph coarsening. The authors first extends the graph coarsening metric Restricted Spectral Approximation (RSA) by Loukas, in order to disentangle the lifting matrix and the reduction matrix. By observing the additional flexibility in defining reduction matrices, the authors proposes three nested sets of reduction matrices to optimize RSA. The authors provide concrete examples of these reduction matrix characterizations via pointers to the existing literature, and propose new choices of reduction matrices. These reduction matrices are compared empirically on the RSA metric and predictive performance of GNN trained on the obtained coarsened graphs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The proposal of a taxonomy of reduction matrices for graph coarsening is interesting and of great interest to the graph learning community.\n\n2. The paper is well written overall, with a clear exposition of the motivation and chosen sets of reduction matrices.\n\nWeakness:\n1. Graph coarsening aims to reduce computation complexity (while maintaining performance). However, for the proposed sets of reduction matrices $E_1, E_2$, they both consist of dense matrices $\\mathbb R^{n \\times N}$. This introduces a heavy memory and time complexity overhead, as shown in Appendix G.2 (e.g., these reduction matrices consists of millions of parameters to be optimized for a medium size graph Cora or Citeseer). The authors do not provide any analysis on complexity and discussion on runtime.\n\n2. The empirical validation is only carried out on two Cora and Citeseer benchmark graphs (with additional synthetic geometric random graphs). The paper can be further strengthened by a more thorough evaluation on more benchmark graphs including both homophilous ones and heterophilous ones (e.g., from [1] or [2], or even random graphs say a two-block stochastic block model where across-block connection probability is higher than within-block probability). \n\nReferences\n[1] Platonov, O., Kuznedelev, D., Diskin, M., Babenko, A., & Prokhorenkova, L. A critical look at the evaluation of GNNs under heterophily: Are we really making progress?. ICLR 2023\n[2] Pei, H., Wei, B., Chang, K. C. C., Lei, Y., & Yang, B. Geom-GCN: Geometric Graph Convolutional Networks. ICLR 2020\n\nQUESTIONS:\n1. For the sparse reduction matrices in set $E_3$, the proposed $P^*_{Q^{\\top}}$ empirically perform quite similar (within standard error bars) to the existing baselines $P_{Loukas}, P_{MP}$. Can the authors discuss their relationship in more details, either conceptually or empirically?\n\n2. The authors discuss in the introduction that \"$P$ is taken as the Moore-Penrose pseudo-inverse...In this paper, we challenge this choice and argue that there is no real reason for it.\". But the characterization of $E_3$ clearly shows the advantages of $P$ being Moree-Penrose pseudo-inverse: (1) it is sparse (share the same support as $Q$); (2) it admits a close-form given $Q$, so no further optimization is needed. From the RSA evaluation, the pseudo-inverse $P$ choice underperforms $P^*_{Q^{\\top}}$ requiring optimization. Do the authors agree? If so, I would suggest reword this discussion.\n\n3. Although the characterization of $E_1, E_2$ reduction matrices is interesting, they introduce heavy computational overhead (Appendix G.2) and seem to bring very little advantage to GNN performance (Table 1). Can the authors discuss possible sparser characterizations or more efficient implementations? \n\nMinor:\na) Definition 2 is the extended version of RSA, which differs from the one proposed by Loukas in disentangling the role of the lifting matrix and the reduction matrix. I would suggest adding the remark before/after this to remind the reader.\n\nb) The notation $\\Delta$ in Lemma 6: does it refer to the $\\Delta$-Laplacian? I didn't find it defining anywhere close to Lemma 6 nor its proof.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper focuses on two detailed issues in graph coarsening: 1) What are the admissible degrees of freedom for the reduction matrix? and 2) Can we improve the RSA by simply modifying the reduction matrix alone? Overall, this paper conducts an in-depth and detailed theoretical study of graph coarsening operations and achieves valuable conclusions. I appreciate the work done in this paper.

STRENGTHS AND WEAKNESSES:
Strengths:

1.This paper is well-written and explains the background and context of the problem in detail.

2.The issues addressed in this paper are highly valuable for the field.

3.The theoretical analysis is solid.

 Weaknesses:

1.However, the experiments in this paper are still insufficient. Although this is a theoretical paper, I still hope the authors can provide more experimental results to support the theoretical conclusions. If the authors further expand the experiments, I will consider increasing my score. Specifically, I would like to see results on at least five datasets, including larger-scale datasets like Arxiv, which current graph coarsening algorithms are capable of handling.

QUESTIONS:
1. Please provide corresponding experiments to support the theoretical analysis. The specific experimental issues have already been mentioned in the weaknesses section.

LIMITATIONS:
Yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-05