Recovery Guarantees for Kernel-based Clustering under Non-parametric Mixture Models
This work addresses a gap in theoretical foundations for kernel-based clustering, offering insights for practitioners in machine learning and statistics, though it appears incremental by extending existing theory to less restrictive assumptions.
The paper tackles the lack of statistical guarantees for kernel-based clustering under non-parametric mixture models, providing necessary and sufficient separability conditions for consistent recovery of true clustering and establishing an equivalence between kernel-based and kernel density-based clustering.
Despite the ubiquity of kernel-based clustering, surprisingly few statistical guarantees exist beyond settings that consider strong structural assumptions on the data generation process. In this work, we take a step towards bridging this gap by studying the statistical performance of kernel-based clustering algorithms under non-parametric mixture models. We provide necessary and sufficient separability conditions under which these algorithms can consistently recover the underlying true clustering. Our analysis provides guarantees for kernel clustering approaches without structural assumptions on the form of the component distributions. Additionally, we establish a key equivalence between kernel-based data-clustering and kernel density-based clustering. This enables us to provide consistency guarantees for kernel-based estimators of non-parametric mixture models. Along with theoretical implications, this connection could have practical implications, including in the systematic choice of the bandwidth of the Gaussian kernel in the context of clustering.