  Determinant maximization provides an elegant generalization of problems in
many areas, including convex geometry, statistics, machine learning, fair
allocation of goods, and network design. In an instance of the determinant
maximization problem, we are given a collection of vectors $v_1,\ldots, v_n \in
\mathbb{R}^d$, and the goal is to pick a subset $S\subseteq [n]$ of given
vectors to maximize the determinant of the matrix $\sum_{i \in S} v_iv_i^\top$,
where the picked set of vectors $S$ must satisfy some combinatorial constraint
such as cardinality constraint ($|S| \leq k$) or matroid constraint ($S$ is a
basis of a matroid defined on $[n]$).
  In this work, we give a combinatorial algorithm for the determinant
maximization problem under a matroid constraint that achieves
$O(d^{O(d)})$-approximation for any matroid of rank $r\geq d$. This complements
the recent result of~\cite{BrownLPST22} that achieves a similar bound for
matroids of rank $r\leq d$, relying on a geometric interpretation of the
determinant. Our result matches the best-known estimation
algorithms~\cite{madan2020maximizing} for the problem, which could estimate the
objective value but could not give an approximate solution with a similar
guarantee. Our work follows the framework developed by~\cite{BrownLPST22} of
using matroid intersection based algorithms for determinant maximization. To
overcome the lack of a simple geometric interpretation of the objective when $r
\geq d$, our approach combines ideas from combinatorial optimization with
algebraic properties of the determinant. We also critically use the properties
of a convex programming relaxation of the problem introduced
by~\cite{madan2020maximizing}.
