Title: Graph Persistence goes Spectral
Venue: NeurIPS 2025
Accepted: yes
Keywords: topological data analysis

graph neural networks

graph machine learning
Abstract: Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe --- a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than PH and spectral information on graphs alone. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks.  Code is available at https://github.com/Aalto-QuML/SpectRe/.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, authors propose SpectRe, a novel topological descriptor for graphs that incorporates spectral information into persistent homology (PH) diagrams. SpectRe offers greater expressiveness compared to existing graph descriptors. The authors also present concepts of global and local stability to evaluate existing methods, demonstrating that SpectRe exhibits local stability. Experimental results  showcase the effectiveness of SpectRe and highlight its potential to enhance the performance of graph-based models in various learning tasks.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The authors demonstrate that the expressiveness provided by the non-zero eigenvalues of these spectral descriptors can be equivalently achieved by incorporating the non-zero eigenvalues of the graph Laplacians at all time steps.\n2. The authors conduct further theoretical analysis of SpectRe.\n3. The authors perform experiments, and the results highlight SpectRe's enhanced expressiveness while demonstrating its potential to improve the performance of graph neural networks on related tasks.\n\nWeaknesses:\n1. Overall, this is a strong paper. However, the introduction could be slightly improved by being more concise and clearly articulating the motivation and contributions, particularly for researchers without a background in topological data analysis (TDA).\n2. The work heavily relies on results from previous studies, which limits the novelty of its theoretical contributions.\n3. The experiments provided are insufficient to fully support the authors' claims. For instance, it is unclear whether the authors considered comparisons with local persistent homology methods.\n\nQUESTIONS:\n1. In the paper, authors mentioned \"Homology captures harmonic information (in the sense that the kernel of the graph Laplacian corresponds to the 0-th homology), but there is some non-harmonic information we also want to account for...\" I am not so sure if harmonic information is necessary? Could you please explain a bit?\n2. The experiments provided are insufficient to fully support the authors' claims. For instance, it is unclear whether the authors considered comparisons with local persistent homology methods.\n3. Could you please clarify the steps involved in the procedure described in Section 5 (Integration with GNNs)?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new topological descriptor for graphs, SpectRe, which enhances RePHINE diagrams by incorporating the nonzero eigenvalues of the graph Laplacian at each filtration step. The authors show that SpectRe is strictly more expressive than both RePHINE and the Laplacian spectrum alone, and establish that it is invariant under graph isomorphisms. They also define custom bottleneck-like metrics under which RePHINE is globally stable and SpectRe is locally stable (under an injectivity assumption). Experimental results on graph isomorphism tasks (Cayley graphs, BREC) and real-world GNN benchmarks (MUTAG, ZINC, etc.) show that SpectRe outperforms prior persistence-based descriptors in both discriminative and predictive settings.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\n1. The idea is conceptually clean and fills a real gap in the PH literature. Augmenting persistence diagrams with spectral information has been discussed before, but this is the first implementation that (i) actually improves expressivity over RePHINE and Laplacian individually and (ii) is formalized with a proper stability framework.\n\n2. The theoretical results are solid. The expressivity separation is clearly demonstrated (Figure 3), and the stability results (global for RePHINE, local for SpectRe) are carefully stated and proven with new metrics defined for each case. The counterexample in Example 4.6 is also good for setting expectations.\n\n## Weaknesses\n\n1. The method is significantly more expensive than standard PH descriptors, especially on large or dense graphs. The authors mention scheduling and power methods, but they don’t provide any concrete timing comparisons or resource benchmarks.\n\n2. I do not find the downstream empirical performance on real data to be particularly convincing. For most of the datasets in Table 3, the separation between SpectRe and RePHINE does not seem to be statistically significant. The mean improves slightly, but the standard deviation is rather high in most cases.\n\n3. The stability of SpectRe only holds locally under injectivity, which is a strong assumption. Many real-world filters (e.g., curvature-based or learned ones) may not satisfy this. It would be good to have some robustness analysis or at least an empirical sanity check here.\n\n4. The lack of ablation is noticeable. It’s not clear whether the gains come primarily from the spectral component or whether there is some synergy with RePHINE. A comparison with a \"spectrum-only\" variant would help clarify this.\n\n## Verdict\n\nThis paper introduces a well-motivated and clearly beneficial modification of persistence-based graph descriptors. The improvements over RePHINE and the Laplacian spectrum are theoretically well-justified. That being said, Table 3 does not leave me with a particularly convincing impression of the real-world empirical utility of the method and I do think the lack of ablations and runtime analysis weakens the empirical section. Additionally, the local-only stability is a limitation worth highlighting. All things considered, I am inclined to give the paper a weak reject rating in its current state.\n\nQUESTIONS:\n1. How sensitive is SpectRe to noise or perturbations in the filtration functions, particularly when injectivity is violated?\n\n2. Can the stability results be extended to include non-injective filtrations in some approximate sense?\n\n3. Would using approximate spectral methods (e.g., Nyström) change expressivity, or just reduce fidelity?\n\nLIMITATIONS:\nThe authors note the computational complexity and the lack of global stability for SpectRe. I’d add that the injectivity assumption may be hard to guarantee in real applications, and there’s some ambiguity about how robust SpectRe is to near-ties or small perturbations in filtration values. The method also assumes access to full eigenvalue spectra (at least at small graph sizes), which may limit applicability at larger scale.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper extends the existing work in the line of using Persistent Homology (PH) methods for Graph Representation Learning. \n\nMore specifically, the authors extend the existing work of Immonen et. al. [28] by incorporating Laplacian spectral information into the PH invariants, which are then fed into the filtration analysis typical of PH-methods for GNNs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The exposition is very clear. \n2. The stability analysis is quite thorough. \n\nWeakness\n\n1. The experimental set-up is mainly designed to compare the results of this paper vs the Immomen paper. There are no comparisons with other state-of-the-art GNNs. In particular, I would like to see a baseline of GNNs which use spectral information in a simple manner (there are tons of models, message-passing based or transformed based which do this). The theoretical results (Theorem 3.2, Theorem 3.3) do not seem very hard to prove. \n\nOverall, the paper seems to be focused on exceeding the results in the Immomen paper. Perhaps PH enthusiasts can better describe the significance of these results (stability section): The broader significance of these results is not so clear to me.\n\n2. I do not see how the stability section is connected with the rest of the paper. Why are there no experimental results to demonstrate the significance of these results? \n\n3. The computational costs of SpectRe are very prohibitive, compared to SOTA GNNs.\n\nQUESTIONS:\n\"189 The intuition behind why RePHINE cannot differentiate the\n190 examples in Figure 3 is that it has the same amount of expres191 sive power as counting the number of connected components\n192 and independent cycles (i.e. the harmonic components of the\n193 Laplacian) on a monochromatic graph .. \"\n\nPlease cite this as a theorem from an earlier paper, or prove it somewhere.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces SpectRe, a topological descriptor for graphs that enhances persistent homology (PH) diagrams with spectral information. Designed to be more expressive than prior methods like RePHINE, SpectRe augments persistence tuples with eigenvalues of the graph Laplacian at critical filtration steps. The authors provide a theoretical analysis of the descriptor's expressivity and local stability, and demonstrate its effectiveness on synthetic isomorphism tests and real-world graph learning tasks.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n- The paper is theoretically sound and provides a comprehensive analysis of the proposed SpectRe descriptor. The expressivity proofs and the stability property of are well-executed.\n- The isomorphism tests on synthetic datasets are well-designed and clearly support the theoretical claims about the superior expressive power of SpectRe.\n- The paper is well-written and presents its complex ideas in a structured and understandable manner.\n\n**Weaknesses:**\n- The core idea is to add spectral features to an existing descriptor (RePHINE). This is a natural follow-up step rather than a fundamentally new approach. The method essentially decorates an existing descriptor with a well-known useful feature type. The main theoretical result—that adding more information (spectral features) increases the model's ability to distinguish non-isomorphic graphs—is not surprising. \n- While theoretically more expressive, the empirical improvements on several real-world datasets (Table 3) are marginal. This raises questions about the practical utility of the proposed method, especially when considering its significant computational overhead.\n\nQUESTIONS:\n1. **On the Choice of Persistent Laplacian Construction:** The paper argues that persistent Laplacians offer no more expressive power than the standard graph Laplacian. However, the cited work [49] is one of several constructions. How does this result hold up when considering other definitions, such as the one proposed by Mémoli, Wan, and Wang in \"Persistent Laplacians: properties, algorithms and implications\"?\n2. **On Comparison with Multi-Parameter Persistence:** The paper builds upon single-parameter persistence. How do you see SpectRe's expressivity in relation to representations derived from Multi-Parameter Persistence (MPH) [1, 2, 3]? Since MPH can capture more complex interactions by using multiple simultaneous filtrations, could replacing the single-parameter PH backbone of SpectRe with an MPH backbone lead to an even more powerful descriptor?\n3. **On Local vs. Global Stability:** Could you provide more intuition on the practical implications of local stability? Are there realistic scenarios or types of graph data where you would expect the lack of global stability to be a significant issue?\n4. **On the Choice of Filtration:** The performance of SpectRe can be dependent on the choice of the initial vertex and edge filtration functions. Do you have any general guidelines or heuristics for choosing effective filtration functions for a given graph learning task?\n\n\n**References**\n\n[1] Magnus Bakke Botnan and Michael Lesnick. An introduction to multiparameter persistence. In Representations of algebras and related structures, EMS Ser. Congr. Rep., pages 77–150. EMS Press, Berlin, [2023] ©2023.\n\n[2] Oliver Vipond. Multiparameter persistence landscapes. Journal of Machine Learning Research, 21(61):1–38, 2020.\n\n[3] Xin, C., Mukherjee, S., Samaga, S. N., and Dey, T. K. GRIL: a 2-parameter persistence based vectorization for machine learning. In Proceedings of 2nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML), volume 221 of Proceedings of Machine Learning Research, pp. 313–333. PMLR, 7 2023.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-06"}