  Randomized Numerical Linear Algebra (RandNLA) uses randomness to develop
improved algorithms for matrix problems that arise in scientific computing,
data science, machine learning, etc. Determinantal Point Processes (DPPs), a
seemingly unrelated topic in pure and applied mathematics, is a class of
stochastic point processes with probability distribution characterized by
sub-determinants of a kernel matrix. Recent work has uncovered deep and
fruitful connections between DPPs and RandNLA which lead to new guarantees and
improved algorithms that are of interest to both areas. We provide an overview
of this exciting new line of research, including brief introductions to RandNLA
and DPPs, as well as applications of DPPs to classical linear algebra tasks
such as least squares regression, low-rank approximation and the Nystr\"om
method. For example, random sampling with a DPP leads to new kinds of unbiased
estimators for least squares, enabling more refined statistical and inferential
understanding of these algorithms; a DPP is, in some sense, an optimal
randomized algorithm for the Nystr\"om method; and a RandNLA technique called
leverage score sampling can be derived as the marginal distribution of a DPP.
We also discuss recent algorithmic developments, illustrating that, while not
quite as efficient as standard RandNLA techniques, DPP-based algorithms are
only moderately more expensive.
