  Changepoints are a very common feature of Big Data that arrive in the form of
a data stream. In this paper, we study high-dimensional time series in which,
at certain time points, the mean structure changes in a sparse subset of the
coordinates. The challenge is to borrow strength across the coordinates in
order to detect smaller changes than could be observed in any individual
component series. We propose a two-stage procedure called `inspect' for
estimation of the changepoints: first, we argue that a good projection
direction can be obtained as the leading left singular vector of the matrix
that solves a convex optimisation problem derived from the CUSUM transformation
of the time series. We then apply an existing univariate changepoint estimation
algorithm to the projected series. Our theory provides strong guarantees on
both the number of estimated changepoints and the rates of convergence of their
locations, and our numerical studies validate its highly competitive empirical
performance for a wide range of data generating mechanisms. Software
implementing the methodology is available in the R package
`InspectChangepoint'.
