Spectral Clustering with Side Information
In the graph clustering problem with a planted solution, the input is a graph on π vertices partitioned into π clusters, and the task is to infer the clusters from graph structure. A standard assumption is that clusters induce well-connected subgraphs (i.e. Ξ©β‘(1)-expanders), and connections between clusters are sparse (i.e. clusters form π-sparse cuts). Such a graph defines the clustering uniquely up to βπ misclassification rate, and efficient algorithms for achieving this rate are known. While this vanilla version of graph clustering is extremely well studied, in the practice of graph analysis vertices of the graph are typically equipped with labels, or features, that provide additional information on cluster ids of the vertices. For example, each vertex could be equipped with a cluster label that is corrupted independently with probability πΏ. Using either of the two sources of information separately leads to misclassification rate minβ‘{π,πΏ}, but can one combine the two to achieve misclassification rate βπβ’πΏ? In this paper, we give an affirmative answer to this question, and furthermore show that such a misclassification rate can be achieved in sublinear time in the number of vertices π. Our key algorithmic insight is a new observation on βspectrally ambiguousβ vertices in a well-clusterable graph. While our sublinear-time classifier achieves the nearly optimal βΛπβ‘(πβ’πΏ) misclassification rate, the approximate clusters that it outputs do not necessarily induce expanders in the graph πΊ. In our second result, we give a polynomial-time algorithm for reweighting edges of the input graph from the original (π,π,Ξ©β‘(1)-clusterable instance to a (π,Λπβ‘(πβ’πΏ),Ξ©β‘(1))-clusterable instance (for constant π), improving sparsity of cuts nearly optimally and preserving expansion properties of the communities β an algorithm for refining community structure of the input graph.
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Fichtenberger, H., Kapralov, M., Kochetkova, E., Lattanzi, S., Mazzali, D., & Wrzos-Kaminska, W. (2026). Spectral Clustering with Side Information. In Proceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (pp. 4327β4341). Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611978971.159
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EPFL
Γcole Polytechnique FΓ©dΓ©rale de Lausanne
EPFL
2026-01
Philadelphia, PA
9781611978971
4327
4341
REVIEWED
EPFL