  We consider a streaming data model in which n sensors observe individual
streams of data, presented in a turnstile model. Our goal is to analyze the
singular value decomposition (SVD) of the matrix of data defined implicitly by
the stream of updates. Each column i of the data matrix is given by the stream
of updates seen at sensor i. Our approach is to sketch each column of the
matrix, forming a "sketch matrix" Y, and then to compute the SVD of the sketch
matrix. We show that the singular values and right singular vectors of Y are
close to those of X, with small relative error. We also believe that this bound
is of independent interest in non-streaming and non-distributed data collection
settings.
  Assuming that the data matrix X is of size Nxn, then with m linear
measurements of each column of X, we obtain a smaller matrix Y with dimensions
mxn. If m = O(k \epsilon^{-2} (log(1/\epsilon) + log(1/\delta)), where k
denotes the rank of X, then with probability at least 1-\delta, the singular
values \sigma'_j of Y satisfy the following relative error result
  (1-\epsilon)^(1/2)<= \sigma'_j/\sigma_j <= (1 + \epsilon)^(1/2) as compared
to the singular values \sigma_j of the original matrix X. Furthermore, the
right singular vectors v'_j of Y satisfy
  ||v_j-v_j'||_2 <= min(sqrt{2},
(\epsilon\sqrt{1+\epsilon})/(\sqrt{1-\epsilon}) max_{i\neq j}
(\sqrt{2}\sigma_i\sigma_j)/(min_{c\in[-1,1]}(|\sigma^2_i-\sigma^2_j(1+c\epsilon)|)))
as compared to the right singular vectors v_j of X. We apply this result to
obtain a streaming graph algorithm to approximate the eigenvalues and
eigenvectors of the graph Laplacian in the case where the graph has low rank
(many connected components).
