aiqbench analysis
This paper proposes kernelised covariance matrices as an alternative to PCA for dimension reduction and spectral clustering, aimed at Gaussian mixtures with heavily imbalanced cluster sizes. The authors report theoretical bounds and simulations showing better clustering guarantees over a wider range of intra-cluster separations, though the evidence is mainly theoretical and synthetic. For ML readers, the practical question is whether these gains carry over to real, high-dimensional data and to scalable implementations.
Kernelised covariance matrices, dimension reduction and imbalanced spectral clustering
arXiv:2610.33177v1 Announce Type: cross Abstract: The Gaussian Mixture Model (abbrv. GMM) is a classic generative model for clustered data, for which spectral clustering based on dimension reduction via PCA has been an effective framework. In this work, we investigate kernelised covariance matrices as an alternative mechanism for dimension reduction and spectral clustering, focusing on truncation kernels that select pairs of data points with suitable separation constraints. The structure of these matrices is motivated by considerations from statistical mechanics of repulsive particle systems. Our approach is particularly relevant for the setting of imbalanced clusters, where the sizes of the largest and smallest clusters have a diverging ratio with the problem size, and existing PCA-based methods turn out to be of limited effectiveness. We demonstrate efficient dimension reduction and clustering performance of these modified covariance matrices, substantiated by quantitative bounds. In particular, they achieve superior dimension reduction guarantees compared to PCA for a wide range of parameters in the regime of imbalanced clusters. For spectral clustering, our approach exhibits improvements over PCA in the imbalanced setting, in particular guaranteeing accurate clustering in a much larger regime of intra-cluster separations. Our theoretical guarantees reflected in a range of simulation experiments that exhibit these improvements for unbalanced clustering problems. Our analytical approach, based on kernel Laplacians, provide a toolbox as well as a relevant conceptual ingredient for clustering problems at large. We believe that our work will motivate the application of such kernelised covariance matrices for related classes of statistical problems, as well as spur new investigations in the theory of random matrices.
Abstract:The Gaussian Mixture Model (abbrv. GMM) is a classic generative model for clustered data, for which spectral clustering based on dimension reduction via PCA has been an effective framework. In this work, we investigate kernelised covariance matrices as an alternative mechanism for dimension reduction and spectral clustering, focusing on truncation kernels that select pairs of data points with suitable separation constraints. The structure of these matrices is motivated by considerations from statistical mechanics of repulsive particle systems. Our approach is particularly relevant for the setting of imbalanced clusters, where the sizes of the largest and smallest clusters have a diverging ratio with the problem size, and existing PCA-based methods turn out to be of limited effectiveness. We demonstrate efficient dimension reduction and clustering performance of these modified covariance matrices, substantiated by quantitative bounds. In particular, they achieve superior dimension reduction guarantees compared to PCA for a wide range of parameters in the regime of imbalanced clusters. For spectral clustering, our approach exhibits improvements over PCA in the imbalanced setting, in particular guaranteeing accurate clustering in a much larger regime of intra-cluster separations. Our theoretical guarantees reflected in a range of simulation experiments that exhibit these improvements for unbalanced clustering problems. Our analytical approach, based on kernel Laplacians, provide a toolbox as well as a relevant conceptual ingredient for clustering problems at large. We believe that our work will motivate the application of such kernelised covariance matrices for related classes of statistical problems, as well as spur new investigations in the theory of random matrices.
Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR); Spectral Theory (math.SP)
arXiv:2610.33177 [stat.ML]
(or arXiv:2610.33177v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2610.33177 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 356 words