  We consider the problem of finding anomalies in high-dimensional data using
popular PCA based anomaly scores. The naive algorithms for computing these
scores explicitly compute the PCA of the covariance matrix which uses space
quadratic in the dimensionality of the data. We give the first streaming
algorithms that use space that is linear or sublinear in the dimension. We
prove general results showing that \emph{any} sketch of a matrix that satisfies
a certain operator norm guarantee can be used to approximate these scores. We
instantiate these results with powerful matrix sketching techniques such as
Frequent Directions and random projections to derive efficient and practical
algorithms for these problems, which we validate over real-world data sets. Our
main technical contribution is to prove matrix perturbation inequalities for
operators arising in the computation of these measures.
