高斯噪声下奇异向量线性形式的扰动
概率论
2015-06-10 v1
摘要
设A ∈ R m × n A\in\mathbb{R}^{m\times n} A ∈ R m × n r r r A = ∑ k = 1 r σ k ( u k ⊗ v k ) A=\sum_{k=1}^r\sigma_k (u_k\otimes v_k) A = ∑ k = 1 r σ k ( u k ⊗ v k ) { σ k , k = 1 , … , r } \{\sigma_k, k=1,\ldots,r\} { σ k , k = 1 , … , r } A A A u k ∈ R m , v k ∈ R n , k = 1 , … , r u_k\in {\mathbb R}^m, v_k\in {\mathbb R}^n, k=1,\ldots, r u k ∈ R m , v k ∈ R n , k = 1 , … , r A ~ = A + X \tilde{A}=A+X A ~ = A + X A A A X ∈ R m × n X\in\mathbb{R}^{m\times n} X ∈ R m × n X i j ∼ N ( 0 , τ 2 ) X_{ij}\sim\mathcal{N}(0,\tau^2) X ij ∼ N ( 0 , τ 2 ) A ~ = ∑ k = 1 m ∧ n σ ~ k ( u ~ k ⊗ v ~ k ) \tilde{A}=\sum_{k=1}^{m\wedge n}\tilde{\sigma}_k(\tilde{u}_k\otimes\tilde{v}_k) A ~ = ∑ k = 1 m ∧ n σ ~ k ( u ~ k ⊗ v ~ k ) σ ~ 1 ≥ … ≥ σ ~ m ∧ n \tilde{\sigma}_1\geq\ldots\geq\tilde{\sigma}_{m\wedge n} σ ~ 1 ≥ … ≥ σ ~ m ∧ n u ~ k , v ~ k , k = 1 , … , m ∧ n \tilde{u}_k,\tilde{v}_k,k=1,\ldots, m\wedge n u ~ k , v ~ k , k = 1 , … , m ∧ n A A A ⟨ u ~ k , x ⟩ , x ∈ R m \langle \tilde u_k,x\rangle, x\in {\mathbb R}^m ⟨ u ~ k , x ⟩ , x ∈ R m ⟨ v ~ k , y ⟩ , y ∈ R n \langle \tilde v_k,y\rangle, y\in {\mathbb R}^n ⟨ v ~ k , y ⟩ , y ∈ R n O ( log ( m + n ) m ∨ n ) O\biggl(\sqrt{\frac{\log(m+n)}{m\vee n}}\biggr) O ( m ∨ n l o g ( m + n ) ) max 1 ≤ i ≤ m ∣ < u ~ k − 1 + b k u k , e i m > ∣ a n d max 1 ≤ j ≤ n ∣ < v ~ k − 1 + b k v k , e j n > ∣ , \max_{1\leq i\leq m}\big|\big<\tilde{u}_k-\sqrt{1+b_k}u_k,e_i^m\big>\big|\ \ {\rm and} \ \ \max_{1\leq j\leq n}\big|\big<\tilde{v}_k-\sqrt{1+b_k}v_k,e_j^n\big>\big|, 1 ≤ i ≤ m max ⟨ u ~ k − 1 + b k u k , e i m ⟩ and 1 ≤ j ≤ n max ⟨ v ~ k − 1 + b k v k , e j n ⟩ , b k b_k b k u ~ k , v ~ k \tilde u_k, \tilde v_k u ~ k , v ~ k { e i m , i = 1 , … , m } , { e j n , j = 1 , … , n } \{e_i^m,i=1,\ldots,m\}, \{e_j^n,j=1,\ldots,n\} { e i m , i = 1 , … , m } , { e j n , j = 1 , … , n } R m , R n \mathbb{R}^m, {\mathbb R}^n R m , R n
引用
@article{arxiv.1506.02764,
title = {Perturbation of linear forms of singular vectors under Gaussian noise},
author = {Vladimir Koltchinskii and Dong Xia},
journal= {arXiv preprint arXiv:1506.02764},
year = {2015}
}