arXiv paper proposes multi-kernel spectral clustering with exact recovery guarantees

Zeqin Lin, Guangming Pan, Zhixiang Zhang, and Yinbing Zhou have released a paper on arXiv titled "Multi-kernel spectral clustering: Entrywise eigenvector perturbation bounds and exact recovery" (arXiv:2608.08704). The work addresses a common failure mode in kernel spectral clustering: using a single fixed bandwidth often misses when data contains multiple characteristic pairwise-distance scales, especially in high-dimensional settings. Their solution aggregates kernels with several bandwidths, automatically selected as empirical quantiles of pairwise squared distances. This captures relevant distance scales without any prior knowledge of the population-scale information, making the method practical for real-world datasets.

The core contribution is a rigorous theoretical framework. The authors construct a blockwise constant, low-rank approximation to the empirical multi-kernel matrix and prove row-wise ℓ2,∞ perturbation bounds for its leading spectral components and the associated normalized Laplacian. These entrywise bounds offer far more granular control over spectral embeddings than standard global eigenspace perturbation estimates. Under eigenvalue-gap and cluster-separation conditions, they show that applying approximate K-means to the multi-kernel spectral embedding results in exact recovery with high probability. This gives users a mathematically sound, automatic way to cluster complex, heterogeneous high-dimensional data without tuning bandwidths by hand.