## Refined Learning Bounds for Kernel and Approximate $k$-Means

21 May 2021, 20:43 (modified: 21 Jan 2022, 17:18)NeurIPS 2021 SpotlightReaders: Everyone
Keywords: Kernel $k$-means，Risk bound, Kernel methods, Learning theory
TL;DR: In this paper, we study the statistical properties of kernel $k$-means and Nystr\"{o}m-based kernel $k$-means, and obtain optimal clustering risk bounds, which improves on the existing risk bounds.
Abstract: Kernel $k$-means is one of the most popular approaches to clustering and its theoretical properties have been investigated for decades. However, the existing state-of-the-art risk bounds are of order $\mathcal{O}(k/\sqrt{n})$, which do not match with the stated lower bound $\Omega(\sqrt{k/n})$ in terms of $k$, where $k$ is the number of clusters and $n$ is the size of the training set. In this paper, we study the statistical properties of kernel $k$-means and Nystr\"{o}m-based kernel $k$-means, and obtain optimal clustering risk bounds, which improve the existing risk bounds. Particularly, based on a refined upper bound of Rademacher complexity [21], we first derive an optimal risk bound of rate $\mathcal{O}(\sqrt{k/n})$ for empirical risk minimizer (ERM), and further extend it to general cases beyond ERM. Then, we analyze the statistical effect of computational approximations of Nystr\"{o}m kernel $k$-means, and prove that it achieves the same statistical accuracy as the original kernel $k$-means considering only $\Omega(\sqrt{nk})$ Nystr\"{o}m landmark points. We further relax the restriction of landmark points from $\Omega(\sqrt{nk})$ to $\Omega(\sqrt{n})$ under a mild condition. Finally, we validate the theoretical findings via numerical experiments.
Supplementary Material: pdf
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Code: http://www.csie.ntu.edu.tw/~cjlin/libsvm
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