  We develop a sketching algorithm to find the point on the convex hull of a
dataset, closest to a query point outside it. Studying the convex hull of
datasets can provide useful information about their geometric structure and
their distribution. Many machine learning datasets have large number of samples
with large number of features, but exact algorithms in computational geometry
are usually not designed for such setting. Alternatively, the problem can be
formulated as a linear least-squares problem with linear constraints. However,
solving the problem using standard optimization algorithms can be very
expensive for large datasets. Our algorithm uses a sketching procedure to
exploit the structure of the data and unburden the optimization process from
irrelevant points. This involves breaking the data into pieces and gradually
putting the pieces back together, while improving the optimal solution using a
gradient project method that can rapidly change its active set of constraints.
Our method eventually leads to the optimal solution of our convex problem
faster than off-the-shelf algorithms.
