  When faced with a data set too large to be processed all at once, an obvious
solution is to retain only part of it. In practice this takes a wide variety of
different forms, and among them "coresets" are especially appealing. A coreset
is a (small) weighted sample of the original data that comes with the following
guarantee: a cost function can be evaluated on the smaller set instead of the
larger one, with low relative error. For some classes of problems, and via a
careful choice of sampling distribution (based on the so-called "sensitivity"
metric), iid random sampling has turned to be one of the most successful
methods for building coresets efficiently. However, independent samples are
sometimes overly redundant, and one could hope that enforcing diversity would
lead to better performance. The difficulty lies in proving coreset properties
in non-iid samples. We show that the coreset property holds for samples formed
with determinantal point processes (DPP). DPPs are interesting because they are
a rare example of repulsive point processes with tractable theoretical
properties, enabling us to prove general coreset theorems. We apply our results
to both the k-means and the linear regression problems, and give extensive
empirical evidence that the small additional computational cost of DPP sampling
comes with superior performance over its iid counterpart. Of independent
interest, we also provide analytical formulas for the sensitivity in the linear
regression and 1-means cases.
