Title: SpEx: A Spectral Approach to Explainable Clustering
Venue: NeurIPS 2025
Accepted: yes
Keywords: explainable clustering

spectral clustering

interpretable clustering

cheeger
Abstract: Explainable clustering by axis-aligned decision trees was introduced by Moshkovitz et al. (2020) and has gained considerable interest. Prior work has focused on minimizing the price of explainability for specific clustering objectives, lacking a general method to fit an explanation tree to any given clustering, without restrictions. In this work, we propose a new and generic approach to explainable clustering, based on spectral graph partitioning. With it, we design an explainable clustering algorithm that can fit an explanation tree to any given non-explainable clustering, or directly to the dataset itself. Moreover, we show that prior algorithms can also be interpreted as graph partitioning, through a generalized framework due to Trevisan (2013) wherein cuts are optimized in two graphs simultaneously. Our experiments show the favorable performance of our method compared to baselines on a range of datasets.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper presents SpEx, an explainable approach for spectral clustering. Specifically, the paper considers a generic approach that is based on spectral graph clustering paired with an iterative tree construction algorithm. The paper investigates two variants of the approach: SpEx-Clique that can represent any reference clustering using a clique graph and SpEx-kNN that does not require a reference clustering and is based on the nearest neighbours in the dataset. Experiments on eight datasets show that SpEx-Clique outperforms the interpretable baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Interesting topic that has received significant attention recently\n- Strong theoretical basis including Theorem 2.2 and the generalized framework based on non-uniform sparse cuts\n- Experiments show the proposed approach (specifically, SpEx-Clique) outperforms the baselines\n\n----\n\nWeaknesses:\n- The paper claims that the proposed approach \"has the flexibility to produce a tree T with any desired number of leaves, regardless of the number of clusters in the reference clustering” in contrast to approaches like IMM and EMN. It is not entirely clear to me why this is a benefit, as the other approaches can simply increase the number of clusters in the algorithm that produces the reference clustering. In contrast, my understanding of the proposed approach is that it still suffers from the main limitation of IMM and EMN where one leaf corresponds to one cluster hence, adding additional leaves forces the use of additional clusters in the resulting partition.\n\n- Highly-related works are missing from both literature review and as baselines:\n\t* There are existing works highly-related to spectral clustering that are not mentioned or considered as baselines. Specifically, [1] is an optimization-based approach that can support various clustering objectives and includes experiments with both kernel and spectral objectives, [2] which is cited, includes a generic approach that can be used with non-Euclidean objectives and was used Spherical K-Means objective in [2] and with spectral objective in [3]; [3] focused on interpretable clustering with alternative interpretable model (generalized additive model) and has been tested with SpectralNet embeddings; [4] is an interpretable Kernel K-Means approach (based on decision trees) that does not require centroids.\n\t* The widely used ExKMC algorithm [5] (that can utilize multiple leaves per cluster) is cited but is not used as a baseline.\n\n- Experimental results:\n\t* Experiments only compare the approach to the baselines in terms of external evaluation (ARI and AMI are based on labels), however it would be very interesting to report both “the cost of interpretability” in the proposed approach and also the suboptimality as a result of using a greedy optimization procedure.\n\t* It would be very useful to report aggregate metrics across datasets like average ARI and AMI to better understand the performance across datasets, ideally with standard deviation.\n\t* While results show that SpEx-Clique outperforms the baselines, the gains seem somewhat limited and varies across datasets.\n\n- Minor:\n- Line 141: “graph nodes graph”\n- what distance metric is used for the kNN graph?\n\n----\n\n[1] Cohen, E. (2023, May). Interpretable clustering via soft clustering trees. In International Conference on Integration of Constraint Programming, Artificial Intelligence, and Operations Research (pp. 281-298). Cham: Springer Nature Switzerland.\n\n[2] Gabidolla, M., & Carreira-Perpiñán, M. Á. (2022, August). Optimal interpretable clustering using oblique decision trees. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (pp. 400-410).\n\n[3] Upadhya, N., & Cohen, E. (2024, January). NeurCAM: Interpretable Neural Clustering via Additive Models. In ECAI.\n\n[4] Ohl, L., Mattei, P. A., Leclercq, M., Droit, A., & Precioso, F. (2024). Kernel KMeans clustering splits for end-to-end unsupervised decision trees. arXiv preprint arXiv:2402.12232.\n\n[5] Frost, N., Moshkovitz, M., & Rashtchian, C. (2020). ExKMC: Expanding Explainable $ k $-Means Clustering. arXiv preprint arXiv:2006.02399.\n\nQUESTIONS:\nPlease see my main concerns above. No further questions.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper suggests a new approach to explainable clustering with decision trees, based on spectral graph theory. Concretely, the authors propose SPEX, an algorithm that constructs a decision tree based on a graph $G(X,E,w)$. The tree is constructed by iteratively choosing a leaf and an axis-aligned cut that maximize the decrease in conductance at that leaf. The goal is to greedily find a tree $T$ that minimizes a quantity $\\bar{\\Psi}_G(T)$ that can be viewed as a normalized cut objective for multi-way partitions. The paper justifies this choice of objective by adapting a theorem that, broadly speaking, asserts that axis-aligned cuts with low conductance exist if points that are connected in the graph through an edge are on average closer than random points. The SPEX algorithm can be used for any graph, in particular for the clique graph in which points from the same (reference) cluster are connected. Alternatively, one may also directly use the kNN graph as input. Building on this graph characterization of explainable clustering, the paper shows that previous algorithms based on reference clusters (slightly adapted variant of IMM, EMN, CART) can be viewed as minimizing different non-uniform sparse cuts. The paper also includes a set of experiments that measure Adjusted Rand Index (ARI) and Adjusted Mutual Information (AMI) between different explainable clustering methods and ground truth clusters (corresponding to labels).\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\nThis paper is a useful contribution to the literature on explainable clustering via decision trees. Most recent works restrict themselves to studying theoretical approximation guarantees for decision trees in k-means and k-medians (with an extension to kernel k-means with certain kernels). Posing the question in a graph framework is a new perspective. It generalizes some of the previous works, allows insights into what distinguishes some algorithms from others (Section 3 is interesting), and gives a solid motivation for the proposed algorithm. The method can take any graph as input, including kNN which in some applications is more meaningful than e.g. reference cluster centers obtained via k-means. Overall, the paper is written well.\n\n**Weaknesses**\n\nWhile the method itself is theoretically motivated (Theorem 2.2.), the authors do not use this result to give guarantees for their explainable clustering algorithm (beyond a small k-means derivation in Appendix A.2). It should also be noted that there are several papers on clustering with decision trees beyond theoretical studies on price of explainability, some of which also do not need reference centers, and CART is just one of them. For example, see [1,2 and references therein].\n\nMinor comments and typos:\n- The first figure seems to label the points according to the Gaussian they came from. Would it not be more reasonable in clustering to colour them according to some k-means partition?\n- $e_G$ and $vol_G$ not defined.\n- Line 249: graph-theortic\n\n[1] Fraiman, Ricardo, Badih Ghattas, and Marcela Svarc. \"Interpretable clustering using unsupervised binary trees.\" Advances in Data Analysis and Classification 7 (2013): 125-145.\n\n[2] Frost, Nave, Michal Moshkovitz, and Cyrus Rashtchian. \"ExKMC: Expanding Explainable $ k $-Means Clustering.\" arXiv preprint arXiv:2006.02399 (2020).\n\nQUESTIONS:\n- It seems in Appendix Lemma A.10 you are actually also bounding the number of errors of a random cut. Can your analysis be used to say something about the price of Random Cut?\n\n- What is the price of explainability for SPEX on k-means experiments? (I am mainly asking this because I feel it should be included, given that the methods you compare against aim to minimize that quantity).\n\n- Did the authors consider comparing against other practical explainable clustering algorithms [1,2]?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a spectral graph partitioning-based explainable clustering method to overcome the limitations of previous decision tree-based explainable clustering approaches. It primarily introduces two methods: SPEX-Clique and SPEX-kNN. The paper conducts experiments comparing these methods against EMN, CART, and Kernel IMM, and analyzes the implications of the results.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n- The paper clearly analyzes the limitations of past research and effectively highlights the significant importance of explainability.\n\n- It attempts to overcome the limitations of previous studies by basing its approach on spectral graph partitioning.\n\n- The paper presents experimental results across nine datasets, analyzing the strengths and weaknesses of the proposed methods.\n\n\n## Weaknesses\n- The paper does not sufficiently cover explainability aspects from past research. More recent studies on explainable unsupervised learning could be discussed to provide a broader context.\n\n- The explainability of the proposed model is not clearly understood. The paper's discussion of prior limitations is too broad, making it difficult to clearly discern the specific contributions of this work. For instance, it is challenging to understand how the bound in Theorem 2.2 directly translates to explainable clustering.\n\n- The experimental design and analysis regarding explainability are insufficient. A deeper consideration and analysis of the benefits are needed. Currently, the performance results in Figure 3 show mixed outcomes.\n\n- A detailed time complexity analysis is missing. While the efficiency of storing graphs and the sweep-line procedure are mentioned, a formal analysis of the overall time complexity for the proposed algorithms (SPEX-Clique and SPEX-kNN) would enhance the paper.\n\n- The greedy approach used for building the decision tree could be further justified. A discussion on its potential trade-offs or any theoretical guarantees (even if not strict optimality) would be beneficial.\n\n- The paper mentions that some datasets are \"Embedded with CLIP\" or \"Embedded with SBERT\". This crucial information warrants a brief discussion in the experimental setup to explain why these specific embeddings were chosen and their potential impact on the results.\n\n- The impact of the parameter $k$ for SPEX-kNN could be discussed in a more comprehensive manner. While additional results are in Appendix B, a summary of the sensitivity and optimal selection process in the main text would be valuable.\n\nQUESTIONS:\n- Please clarify the explainability of the proposed model in the context of existing XAI methods.\n- The paper primarily evaluates performance using standard clustering metrics (ARI, AMI). While these measure agreement with ground truth, they don't directly assess the quality of the explanation or the interpretability of the generated trees. Please provide a deeper analysis for the explainability.\n- Could the authors provide a more robust justification for the greedy approach?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces SpEx, a method to explain clusters of points in $\\mathbb{R}^d$ using decision trees. SpEx is part of the literature that aims to explain clusters of points using binary decision trees. The main advantage of SpEx is its generality in handling input clusters. While previous methods mostly work for centroid-based clustering methods (e.g., k-means and k-median), SpEx aims to generate binary decision tree-based explanations for generic input clusters. Based on a theoretical observation, the paper proposes two instantiations of SpEx (SpEx-Clique and SpEx-kNN). The paper also performs experimental evaluations of their method against other baselines on a range of datasets.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n* The paper proposes a solution to the interesting problem of explaining clusters using decision trees without strong assumptions about the cluster structure. \n\n* The authors propose a theoretical framework to describe other explainability methods through a similar lens as SpEx. This is an interesting contribution to this literature.\n\n* The paper makes a thorough performance comparison across datasets, but some of the results are in the appendix.\n\n* The paper is well organized and does a good job of introducing the problem of cluster explainability to unfamiliar readers. \n\n### Weaknesses\n\n* The provided instantiations of SpEx don’t have strong performance in scenarios where other methods perform well. This is expected, since SpEx makes fewer assumptions about the problem’s structure, hence not a major weakness. \n\n* There is no discussion on how to select the number of leaves. The proposed algorithms introduce one more parameter, the number of leaves $\\ell$, which the authors set to the number of clusters. Although other methods in the literature employ a similar choice, the generality of SpEx raises questions about whether this is the optimal choice for the number of leaves. The paper presents no experimental or theoretical evidence to support their design choice.  \n\n* Important experiments only appear in the appendix (Tab. 2 and Tab. 3). These are important observations, which are relevant to the main conclusion of the paper, and hence should be brought to the main paper.\n\n* There seems to be a gap in performance in low and high-dimensional input spaces. The authors do not discuss this interesting finding enough in the paper. Is there an explanation for this finding on the style of “the curse of dimensionality”?\n\nQUESTIONS:\n* How sensitive is SpEx’s performance to the choice of the number of leaves $\\ell$? Do the authors have any principled guidelines (empirical or theoretical) for selecting $\\ell$ beyond simply matching the number of clusters?\n\n* What happens if I decide to prune the trees after finding them? Pruning may help with the interpretability of the tree.\n\n* Can you provide more insight or analysis into the observed gap between low- and high-dimensional datasets? In particular, is this some sort of “curse of dimensionality” effect?\n\n* Could you bring Tab. 1 and Tab. 2 into the main text paper?\n\n* Could you add further discussion on the complexity of the algorithms? Perhaps list the theoretical (experimental) complexities in a comparison table?\n\nLIMITATIONS:\nThe authors discuss the limitations of their approach. My only suggestion is the addition of a discussion on the interpretability of decision trees with a large number of leaves (e.g., 100 leaves). Are large decision trees really interpretable?", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-04"}