Title: Spectral Graph Coarsening Using Inner Product Preservation and the Grassmann Manifold
Venue: NeurIPS 2025
Accepted: yes
Keywords: Graph coarsening

Grassmann manifold

Graph signals

Spectral methods
Abstract: We propose a novel functorial graph coarsening method that preserves inner products between node features, a property often overlooked by existing approaches focusing primarily on structural fidelity.
By treating node features as functions on the graph and preserving their inner products, our method retains both structural and feature relationships, facilitating substantial benefits for downstream tasks. 
To formalize this, we introduce the Inner Product Error (IPE), which quantifies how the inner products between node features are preserved. 
Leveraging the underlying geometry of the problem on the Grassmann manifold, we formulate an optimization objective that minimizes the IPE, also for unseen smooth functions. We show that minimizing the IPE improves standard coarsening metrics, and illustrate our method’s properties through visual examples that highlight its clustering ability. Empirical results on benchmarks for graph coarsening and node classification show that our approach outperforms existing state-of-the-art methods.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new graph coarsening method for node-featured graphs. For this, the authors define the Inner-Product Error (IPE) $||X^\\top LX - X_c^\\top L_c X_c ||_F^2$ as a metric that captures how well coarsened graphs preserve the Laplacian-induced inner products of node features. The authors design a minimization objective that couples IPE with a Grassmann-alignment term encouraging coarsening maps to be aligned with the top $k$ Laplacian eigenvectors of the graph, and they propose two coarsening algorithms, the feature-aware INGC and the feature-agnostic SINGC. They then provide some theoretical justification for their choice of objective. Finally, they demonstrate superiority of their method through an empirical study, comparing it to common coarsening methods (local variation neighborhood/edges, feature-based graph coarsening), as well as downstream performance on standard node classification tasks.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n- Incorporating node features into graph coarsening methods is important as it is widely acknowledged that node features can be just as important for downstream tasks as the actual graph structure. To my knowledge, the introduced IPE metric is novel.\n- The empirical results (e.g, Table 1) look compelling, as INGC and SINGC (i.e., even not taking into account node features and using just the Grassmannian term) improve standard coarsening methods more or less across the board.\n\n**Weaknesses:**\n- I feel like the theoretical justification is somewhat limited, as, e.g., Proposition 3.4 assumes *perfectly* $k$-smooth features and a coarsening operator $C$ that *precisely* spans the column space of the top $k$ eigenvectors. Also, technical novelty is rather modest, e.g., Proposition 3.5 is a reduction to Thm 13 of [1].\n- The design and theoretical justifications for the method hinge on spectral smoothness of the node features, which real-world graphs might violate. The authors briefly discuss this in § E of the appendix. While they show that standard datasets like Cora, Citeseer etc. fulfill this approximately, I would assume that this would struggle with more heterophilic graphs where node features would also typically be of high frequency. It could strengthen the work to analyze more thoroughly when/how feature non-smoothness actually degrades the coarsening quality.\n- The presentation and clarity could be improved, as not all details were immediately clear to me. For example, it was not obvious to me from the text description what constraint one would project onto with PSGD in INGC. I am adding a few other (though minor) suggestions to improve the clarity under \"Questions\".\n\n[1] Andreas Loukas. *Graph Reduction with Spectral and Cut Guarantees.* JMLR 20 (2019) 1-42.\n\nQUESTIONS:\n- Regarding the Laplacian regularization term in the objective in Eq. (9), in lines 179-180 you state that this \"guarantees that the coarsened graph $\\mathcal{G}_c$ is connected\". Why enforce connectivity unconditionally? Does your method assume the original graph is connected, and if not, wouldn’t this merge naturally disconnected components and reduce generality? I would appreciate if you could clarify the rationale.\n- In the beginning of § 3.1/Eq. (10), a \"smooth graph signal generation mechanism\" is introduced, but it seems that this is never really mentioned again later, as Assumption 3.3 proceeds with defining such signals simply as being exactly in the column space of the top $k$ eigenvectors. Is there any point in stating it, other than motivating Assumption 3.3?\n- In several of the visualizations (like pages 29/30) it is quite difficult to visually discern which coarsening method performs better. Perhaps including more structured or (close to) planar graphs could help. Also, since one of the method's key selling points is preserving node-feature relationships, a small example that actually displays how features are mapped could greatly improve intuition.\n- A few times you mention your method is \"functorial\". Is this a standard term in this context that I'm not aware of? When skimming the abstract for the first time I thought there might be some connection to category theory, but it seems you are using the term more loosely to mean \"functional\" or \"structure-preserving\". I might suggest tweaking the wording.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-22"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a novel graph coarsening method based on a similarity metric in the Grassmann manifold and the inner product defined in graph theory. The authors analyze the principle of the proposed method solidly, and compare it with the previous method in theory. It provides a geometric view to consider the graph coarsening problem. The extensive experiments show the effectiveness of the proposed INGC and SINGC.\n\nSTRENGTHS AND WEAKNESSES:\nPros:\n1. The view that considers the coarsening problem with the Grassmann manifold has novelty and insight. It proposed a new coarsening metric.\n2. The authors make some reasonable assumptions, making theoretical analysis more solid.\n3. The experiments are detailed and convincing, since the authors analyze the model in detail by ablation study and visualization, even validating the sense of assumption.\n\nCons:\n1. In Assumption 3.3 of the $k$-smooth graph signal, I think there is a typo that $x=c^T U^{(k)}$, the correct formula is $x=U^{(k)}c$, since the vector $c$ may be a column vector. Otherwise the dimension is mismatched, and $x=U^{(k)}(U^{(k)})^Tx$ will not hold in Appendix A.2, line 779.\n2. Although the proposed method is solid in theory, I am concerned about the assumption for the $(n-k)$-connected components. Since for a large-scale graph, the inter-community structure is also important, but the proposed method is based on the situation for $(n-k)$-connected components, which may lose information about inter-community connecting patterns.\n3. I think the usage of the Grassmann manifold in the gradient descent (GD) algorithm has a misunderstanding. I do not find any projection on gradients. Since in Eq. 8, you think the coarsening matrix $C$ is also in the Grassmann manifold, but as you apply GD to the point in the Grassmann manifold, if you do not project gradients, you can not guarantee that the gradient vector is in the tangent space, making the updated $C$ destroy the orthogonal property.\n\nQUESTIONS:\n1. Can you make a further discussion about that if a graph truly has $n-k$ connected components, will the proposed method achieve the zero error theoretically?\n2. The weights in coarsening matrix $C$ are soft or hard, i.e., the element ranges in $[0, 1]$ or $\\\\{0, 1 \\\\}$ ?\n3. Can you discuss how the Grassmann manifold plays a vital role in the proposed methods?\n4. Others refer to the **Weakness**.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work presents a graph coarsening method aimed at preserving inner products between node features. A new Inner Product Error (IPE) metric is introduced to measure how the inner products between graph signals are preserved. New connections between IPE minimization and structural consistency of the coarsened graph are established to confirm the effectiveness of IPE. In experiments, both SBM random graph model and real-world graphs are employed to evaluate the proposed INGC model.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The proposed IPE appears novel.\n2. Comprehensive theoretical analyses are provided to explain why IPE minimization is beneficial.\n3. The paper is well-written and probably easy to follow.\n\nWeaknesses:\n\n1. The overall contribution is somewhat limited. Apart from the IPE term, the rest of the objective is heavily borrowed from FGC [14]. Notably, the Grassmann manifold-based inner product preservation is a commonly used technique in graph signal processing, so I don't consider it a novel contribution.\n2. As described in the first paragraph of the Introduction, the motivation is to speed up graph machine learning by reducing the graph. However, the theoretical time complexity of the proposed method is quadratic in the number of nodes, making it hard to handle large-scale graphs in practice. For small-to-medium scale graphs, there is typically no need to perform graph reduction beforehand. This nature makes the proposed method less meaningful. There are also no experiments conducted on large-scale graphs such as ogbn-products and ogbn-papers100M.\n3. The baselines are outdated.\n\nQUESTIONS:\n1. Is the proposed method applicable to heterophilic graphs where nearby nodes are likely from different classes?\n2. Given a graph Laplacian $L$ and graph signals $X$, Dirichlet Energy considers only the diagonal elements of $X^TLX$, i.e. $tr(X^TLX)$, while the proposed IPE additionally considers off-diagonal elements. **In addition to existing theoretical results in Prop. 3.4**, how to understand the contributions of these off-diagonal elements?\n3. How to handle graphs without node attributes?\n\nLIMITATIONS:\nscalablity and applicability to graphs of varying homophily could be discussed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes the inner product error as a loss function for evaluating the quality of a graph coarsening. In particular, for a graph Laplacian $L$ with node features $X$ and a coarsened Laplacian $L_c$ with coarsened features $X_c$, the proposed inner product error is defined $\\| X^TLX - X_c^TL_cX \\|_F^2$. The authors further propose the INGC algorithm for graph coarsening. INGC finds a coarsening by solving a constrained optimization problem (Equations 8 and 9). The loss function includes the inner product error between the original graph and the coarsened graph, as well as smoothness conditions on the coarsening and regularization terms aimed at making the coarsening well-defined and the coarsened graph connected. The paper also presents some theoretical justfications of the inner-product loss, as well as experiments on several benchmark datasets.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n- The inner-product loss makes a lot of sense. The authors use theoretical results to justify the two main components of their loss terms. Propostions 3.5 show that minimizing the proposed loss also minimizes the losses REE and RE (definitions 2.1 and 2.2), two existing losses from the graph coarsening literature. While Proposition 3.2 and 3.4 don't hold in all cases, Proposition 3.5 provides an analysis that works in all cases and that gives better theoretical results the lower the loss. Moreover, the authors perform a simple experiment (table 5) to show the conditions of Proposition 3.5 are realistic.\n\n- The proposed coarsenings outperform existing coarsening techniques on several datasets and on several evaluation metrics (Table 1). \n\n\n## Weaknesses  \n\n- While the proposed inner-product loss is new and well-motivated, its contribution appears a llittle incremental. For the theoretical results, the conditions for Proposition 3.2 are too strong, and Proposition 3.4 is not surprising.  (Proposition 3.5 is the most interesting result of the paper in my view.) \n\n- While Table 1 shows better graph coarsening metrics of proposed methods, the advantage in node classification is not large (Table 2). At the same, the time to compute INGC and SINGC can be quite a bit slower than existing method FGC – see Table 4 in the Supplement – even though they have similar asymptotic runtimes.\n\nQUESTIONS:\n## Questions\n\n(1) The statement of Proposition 3.2 requires a certain condition (the graph has n-k connected components), but I cannot see where this condition is used in the proof. Why is this condition needed?\n\n(2) Does the runtime of INGC in Table 3 in Supplement account for the time to do the projection?\n\n## Suggestions\n\n- I believe for the properties of Equation (7) to hold, it needs to be the case that P_{i,j} > 0 if *and only if* j is mapped to the ith super node above. The way it is written in your paper, only one of these conditions is required for P. (I am basing this off the proof of Proposition 6 in [Loukas, 2020])\n\n- The proof environments seem a little weird. I would suggest to use the \\begin{proof}\\end{proof} commands that are a part of amsthm.\n\n## Typos\n\n- Line 148: rank(L) < k\n\n- Proposition 3.2: The phrasing \"for any two graph signals\" is ambiguous, as \"for any\" could either mean for all or there exists. I would recommend changing this to \"for all pairs of graph signals\".\n\n- Proposition 3.2: In this main text, this result is called a \"Proposition\", but in the appendix, it is called a \"Theorem\". Likewise, Line 209 refers to Proposition 3.4 as \"Theorem 3.4\".\n\n- Appendix, Line 818: The LaTeX commands \\log and \\det should be used.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}