  We study the problem of estimating multivariate log-concave probability
density functions. We prove the first sample complexity upper bound for
learning log-concave densities on $\mathbb{R}^d$, for all $d \geq 1$. Prior to
our work, no upper bound on the sample complexity of this learning problem was
known for the case of $d>3$. In more detail, we give an estimator that, for any
$d \ge 1$ and $\epsilon>0$, draws $\tilde{O}_d \left( (1/\epsilon)^{(d+5)/2}
\right)$ samples from an unknown target log-concave density on $\mathbb{R}^d$,
and outputs a hypothesis that (with high probability) is $\epsilon$-close to
the target, in total variation distance. Our upper bound on the sample
complexity comes close to the known lower bound of $\Omega_d \left(
(1/\epsilon)^{(d+1)/2} \right)$ for this problem.
