Perturbation of Linear Forms of Singular Vectors Under Gaussian Noise

Let A is an element of R-mxn be a matrix of rank r with singular value decomposition (SVD) A = Sigma(r)(k=1) sigma(k)(u(k) circle times nu(k)) where {sigma(k) , k = 1, . . . , r} are singular values of A (arranged in a non-increasing order) and u(k) is an element of R-m; nu(k) is an element of R-n , k = 1, . . . , r are the corresponding left and right orthonormal singular vectors. Let (A) over tilde = A + X be a noisy observation of A where X is an element of R-mxn is a random matrix with i.i.d. Gaussian entries, X-ij similar to N (0, tau(2)) and consider its SVD (A) over tilde = Sigma(m Lambda n)(k=1) (sigma) over tilde (k)((u) over tilde (k) circle times (nu) over tilde (k)) with singular values (sigma) over tilde (1) >= . . . >= (sigma) over tilde (m Lambda n) and singular vectors (u) over tilde (k,) (nu) over tilde (k) , k = 1, . . . , m Lambda n. The goal of this paper is to develop sharp concentration bounds for linear forms <(u) over tilde (k) , x >, x is an element of R-m and <(nu) over tilde (k) , y >, y is an element of 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 O (root log(m+n)/ mVn ) (holding with a high probability) on max(1 <= i <=) (m) vertical bar <(u) over tilde (k) - root 1 + b(k)u(k), e(i)(m)>vertical bar and max(1 <=) j (<=) (n) vertical bar <(nu) over tilde (k) - root 1 + b(k)nu(k), e(j)(n)>vertical bar where b(k) are properly chosen constants characterizing the bias of empirical singular vectors (u) over tilde (k) , (nu) over tilde (k) and {e(i)(m) , i = 1, . . . , m}, {e(j)(m), j = 1, . . . , n} are the canonical bases of R-m, R-n, respectively.