Is $k \times k$ Matrix Eigendecomposition Sufficient for Spectral Clustering?
Keywords: Spectral clustering, Kernel mean embedding, Matrix eigendecomposition
Abstract: Spectral clustering has been widely used in clustering tasks due to its effectiveness. However, its key step, eigendecomposition of an $n\times n$ matrix, is computationally expensive for large-scale datasets. Recent works have proposed methods to reduce this complexity, such as Nystr\"om method approximation and landmark-based approaches. While these methods aim to maintain good clustering quality while performing eigendecomposition on smaller matrix. The minimum matrix size required for spectral decomposition in spectral clustering is $k\times k$ (where $k$ is the number of clusters), as it needs to obtain $n\times k$ k-dimensional spectral embedding features. However, no algorithm can achieve good clustering performance with only a $k\times k$ matrix eigendecomposition currently. In this paper, we propose a novel distribution-based spectral clustering. Our method constructs an $n\times k$ bipartite graph between n data points and k distributions, enabling the eigendecomposition of only a $k\times k$ matrix while preserving clustering quality. We demonstrate that our approach can achieve efficient and effective spectral clustering through $k\times k $matrix eigendecomposition.
Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
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Submission Number: 7029
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