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Gain with no Pain: Efficiency of Kernel-PCA by Nyström Sampling

Sterge, N; Sriperumbudur, B. K.; Rosasco, L.; and Rudi, A. (August 2020, Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics)

In this paper, we analyze a Nyström based approach to efficient large scale kernel principal component analysis (PCA). The latter is a natural nonlinear extension of classical PCA based on considering a nonlinear feature map or the corresponding kernel. Like other kernel approaches, kernel PCA enjoys good mathematical and statistical properties but, numerically, it scales poorly with the sample size. Our analysis shows that Nyström sampling greatly improves computational efficiency without incurring any loss of statistical accuracy. While similar effects have been observed in supervised learning, this is the first such result for PCA. Our theoretical findings are based on a combination of analytic and concentration of measure techniques. Our study is more broadly motivated by the question of understanding the interplay between statistical and computational requirements for learning.

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Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of the characteristic property of the tensor product kernel.