  We study private matrix analysis in the sliding window model where only the
last $W$ updates to matrices are considered useful for analysis. We give first
efficient $o(W)$ space differentially private algorithms for spectral
approximation, principal component analysis, and linear regression. We also
initiate and show efficient differentially private algorithms for two important
variants of principal component analysis: sparse principal component analysis
and non-negative principal component analysis. Prior to our work, no such
result was known for sparse and non-negative differentially private principal
component analysis even in the static data setting. These algorithms are
obtained by identifying sufficient conditions on positive semidefinite matrices
formed from streamed matrices. We also show a lower bound on space required to
compute low-rank approximation even if the algorithm gives multiplicative
approximation and incurs additive error. This follows via reduction to a
certain communication complexity problem.
