  Dimensionality reduction is a first step of many machine learning pipelines.
Two popular approaches are principal component analysis, which projects onto a
small number of well chosen but non-interpretable directions, and feature
selection, which selects a small number of the original features. Feature
selection can be abstracted as a numerical linear algebra problem called the
column subset selection problem (CSSP). CSSP corresponds to selecting the best
subset of columns of a matrix $X \in \mathbb{R}^{N \times d}$, where
\emph{best} is often meant in the sense of minimizing the approximation error,
i.e., the norm of the residual after projection of $X$ onto the space spanned
by the selected columns. Such an optimization over subsets of $\{1,\dots,d\}$
is usually impractical. One workaround that has been vastly explored is to
resort to polynomial-cost, random subset selection algorithms that favor small
values of this approximation error. We propose such a randomized algorithm,
based on sampling from a projection determinantal point process (DPP), a
repulsive distribution over a fixed number $k$ of indices $\{1,\dots,d\}$ that
favors diversity among the selected columns. We give bounds on the ratio of the
expected approximation error for this DPP over the optimal error of PCA. These
bounds improve over the state-of-the-art bounds of \emph{volume sampling} when
some realistic structural assumptions are satisfied for $X$. Numerical
experiments suggest that our bounds are tight, and that our algorithms have
comparable performance with the \emph{double phase} algorithm, often considered
to be the practical state-of-the-art. Column subset selection with DPPs thus
inherits the best of both worlds: good empirical performance and tight error
bounds.
