Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data
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Limiting Behavior of Largest Entry of Random Tensor Constructed by High-Dimensional Data Tiefeng Jiang1 · Junshan Xie2 Received: 19 August 2019 / Revised: 5 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let X k = (xk1 , . . . , xkp ) , k = 1, . . . , n, be a random sample of size n coming from a p-dimensional population. For a fixed integer m ≥ 2, consider a hypercubic random tensor T of mth order and rank n with T=
n k=1
X ⊗ · · · ⊗ Xk = k multiplicity m
n
xki1 xki2 · · · xkim
k=1
1≤i 1 ,...,i m ≤ p
.
Let Wn be the largest off-diagonal entry of T. We derive the asymptotic distribution of Wn under a suitable normalization for two cases. They are the ultra-high-dimension case with p → ∞ and log p = o(n β ) and the high-dimension case with p → ∞ and p = O(n α ) where α, β > 0. The normalizing constant of Wn depends on m and the limiting distribution of Wn is a Gumbel-type distribution involved with parameter m. Keywords Tensor · Extreme-value distribution · High-dimensional data · Stein–Chen Poisson approximation Mathematics Subject Classification (2010) 60F05 · 62H10
Jiang’s research was supported by NSF Grants No. DMS-1406279 and DMS-1916014, and Xie’s research was supported by NNSF Grant No. 11401169 and KRPH Grant No. 20A110001.
B
Junshan Xie [email protected] Tiefeng Jiang [email protected]
1
School of Statistics, University of Minnesota, 224 Church Street, S. E., Minneapolis, MN 55455, USA
2
School of Mathematics and Statistics, Henan University, Kaifeng 475000, China
123
Journal of Theoretical Probability
1 Introduction Let p ≥ 2 be an integer and X ∈ R p be a random vector. The distribution of X serves as a population distribution. Let X k = (xk1 , . . . , xkp ) , 1 ≤ k ≤ n, be a random sample of size n from the population distribution generated by X , that is, X , X 1 , . . . , X n are independent random vectors with a common distribution. The data matrix X = (xki )1≤k≤n,1≤i≤ p produces a hypercubic random tensor T ∈ R p×···× p with order m and rank n defined by T=
n k=1
X ⊗ · · · ⊗ Xk = k
n
m multi ple
xki1 xki2 · · · xkim
k=1
1≤i 1 ,...,i m ≤ p
.
(1)
Researchers have obtained some limiting properties of the tensor defined in (1). By using similar techniques to those in the random matrix theory, Ambainis and Harrow [1] obtained a limiting property of the largest eigenvalue and the limiting spectral distribution of random tensors. Tieplova [18] studied the limiting spectral distribution of the sample covariance matrices constructed by the random tensor data. Lytova [15] further considered the central limit theorem for linear spectral statistics of the sample covariance matrices constructed by the random tensor data. Shi et al. [17] applied limiting properties of the random tensors to an anomaly detection problem in the distribution networks. In this paper, we will study the behavior of the largest off-diagonal entry of the random tensor T when both n and p tend to infinity. Precisely, we will work on the
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