Let $${X}_{k}=(x_{k1}, \ldots , x_{kp})', k=1,\ldots ,n$$ X k = ( x k 1 , … , x kp ) ′ , k = 1 , … , n , be a random sample of size n coming from a p -dimensional population. For a fixed integer $$m\ge 2$$ m ≥ 2 , consider a hypercubic random tensor $$\mathbf {{T}}$$ T of m th order and rank n with $$\begin{aligned} \mathbf {{T}}= \sum _{k=1}^{n}\underbrace{{X}_{k}\otimes \cdots \otimes {X}_{k}}_{\mathrm{multiplicity}\ m}=\Big (\sum _{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots x_{ki_{m}}\Big )_{1\le i_{1},\ldots , i_{m}\le p}. \end{aligned}$$ T = ∑ k = 1 n X k ⊗ ⋯ ⊗ X k ⏟ multiplicity m = ( ∑ k = 1 n x k i 1 x k i 2 ⋯ x k i m ) 1 ≤ i 1 , … , i m ≤ p . Let $$W_n$$ W n be the largest off-diagonal entry of $$\mathbf {{T}}$$ T . We derive the asymptotic distribution of $$W_n$$ W n under a suitable normalization for two cases. They are the ultra-high-dimension case with $$p\rightarrow \infty $$ p → ∞ and $$\log p=o(n^{\beta })$$ log p = o ( n β ) and the high-dimension case with $$p\rightarrow \infty $$ p → ∞ and $$p=O(n^{\alpha })$$ p = O ( n α ) where $$\alpha ,\beta >0$$ α , β > 0 . The normalizing constant of $$W_n$$ W n depends on m and the limiting distribution of $$W_n$$ W n is a Gumbel-type distribution involved with parameter m .

中文翻译：

---

高维数据构造的随机张量最大输入的限制行为

让 $${X}_{k}=(x_{k1}, \ldots , x_{kp})', k=1,\ldots ,n$$ X k = ( xk 1 , … , x kp ) ′ , k = 1 , … , n ，是来自 p 维总体的大小为 n 的随机样本。对于固定整数 $$m\ge 2$$ m ≥ 2 ，考虑一个超三次随机张量 $$\mathbf {{T}}$$ T 的 m 阶并使用 $$\begin{aligned} \mathbf 对 n 进行排序{{T}}= \sum _{k=1}^{n}\underbrace{{X}_{k}\otimes \cdots \otimes {X}_{k}}_{\mathrm{multiplicity}\ m}=\Big (\sum _{k=1}^{n} x_{ki_{1}}x_{ki_{2}}\cdots x_{ki_{m}}\Big )_{1\le i_ {1},\ldots , i_{m}\le p}。\end{aligned}$$ T = ∑ k = 1 n X k ⊗ ⋯ ⊗ X k ⏟ 多重性 m = ( ∑ k = 1 nxki 1 xki 2 ⋯ xkim ) 1 ≤ i 1 , … , im ≤ p 。令 $$W_n$$ W n 是 $$\mathbf {{T}}$$ T 的最大非对角线条目。我们在两种情况下在合适的归一化下推导出 $$W_n$$W n 的渐近分布。它们是具有 $$p\rightarrow \infty $$ p → ∞ 和 $$\log p=o(n^{\beta })$$ log p = o ( n β ) 和$$p\rightarrow \infty $$ p → ∞ 和 $$p=O(n^{\alpha })$$ p = O ( n α ) 其中 $$\alpha ,\beta >0 的高维情况$$ α , β > 0 。$$W_n$$ W n 的归一化常数取决于 m 并且 $$W_n$$ W n 的极限分布是与参数 m 相关的 Gumbel 型分布。