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تسجيل مستخدم جديدPerturbation of linear forms of singular vectors under Gaussian noise
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Let $Ainmathbb{R}^{mtimes n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=sum_{k=1}^rsigma_k (u_kotimes v_k),$ where ${sigma_k, k=1,ldots,r}$ are singular values of $A$ (arranged in a non-increasing order) and $u_kin {mathbb R}^m, v_kin {mathbb R}^n, k=1,ldots, r$ are the corresponding left and right orthonormal singular vectors. Let $tilde{A}=A+X$ be a noisy observation of $A,$ where $Xinmathbb{R}^{mtimes n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij}simmathcal{N}(0,tau^2),$ and consider its SVD $tilde{A}=sum_{k=1}^{mwedge n}tilde{sigma}_k(tilde{u}_kotimestilde{v}_k)$ with singular values $tilde{sigma}_1geqldotsgeqtilde{sigma}_{mwedge n}$ and singular vectors $tilde{u}_k,tilde{v}_k,k=1,ldots, mwedge n.$ The goal of this paper is to develop sharp concentration bounds for linear forms $langle tilde u_k,xrangle, xin {mathbb R}^m$ and $langle tilde v_k,yrangle, yin {mathbb R}^n$ of the perturbed (empirical) singular vectors in the case when the singular values of $A$ are distinct and, more generally, concentration bounds for bilinear forms of projection operators associated with SVD. In particular, the results imply upper bounds of the order $Obiggl(sqrt{frac{log(m+n)}{mvee n}}biggr)$ (holding with a high probability) on $$max_{1leq ileq m}big|big<tilde{u}_k-sqrt{1+b_k}u_k,e_i^mbig>big| {rm and} max_{1leq jleq n}big|big<tilde{v}_k-sqrt{1+b_k}v_k,e_j^nbig>big|,$$ where $b_k$ are properly chosen constants characterizing the bias of empirical singular vectors $tilde u_k, tilde v_k$ and ${e_i^m,i=1,ldots,m}, {e_j^n,j=1,ldots,n}$ are the canonical bases of $mathbb{R}^m, {mathbb R}^n,$ respectively.
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