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The smallest eigenvalues of random kernel matrices: Asymptotic results on the min kernel

Author

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  • Huang, Lu-Jing
  • Liao, Yin-Ting
  • Chang, Lo-Bin
  • Hwang, Chii-Ruey

Abstract

This paper investigates asymptotic properties of the smallest eigenvalue of the random kernel matrix Mn=1nk(Xi,Xj)i,j=1n, where k(x,y)=min{x,y} is the min kernel function and X1,X2,…,Xn are i.i.d. random variables in [0,1]. We prove that under certain conditions, the smallest eigenvalue converges in L1 to zero with the rate of convergence O(n−3). In addition, if the underlying distribution of Xi’s has a bounded density, the distribution of the smallest eigenvalue scaled by n3 converges to an exponential distribution.

Suggested Citation

  • Huang, Lu-Jing & Liao, Yin-Ting & Chang, Lo-Bin & Hwang, Chii-Ruey, 2019. "The smallest eigenvalues of random kernel matrices: Asymptotic results on the min kernel," Statistics & Probability Letters, Elsevier, vol. 148(C), pages 23-29.
    • Handle: RePEc:eee:stapro:v:148:y:2019:i:c:p:23-29
    DOI: 10.1016/j.spl.2018.12.008
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    References listed on IDEAS

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    1. Chang, Lo-Bin & Bai, Zhidong & Huang, Su-Yun & Hwang, Chii-Ruey, 2013. "Asymptotic error bounds for kernel-based Nyström low-rank approximation matrices," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 102-119.
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