  A novel computational approach to log-concave density estimation is proposed.
Previous approaches utilize the piecewise-affine parametrization of the density
induced by the given sample set. The number of parameters as well as non-smooth
subgradient-based convex optimization for determining the maximum likelihood
density estimate cause long runtimes for dimensions $d \geq 2$ and large sample
sets. The presented approach is based on mildly non-convex smooth
approximations of the objective function and \textit{sparse}, adaptive
piecewise-affine density parametrization. Established memory-efficient
numerical optimization techniques enable to process larger data sets for
dimensions $d \geq 2$. While there is no guarantee that the algorithm returns
the maximum likelihood estimate for every problem instance, we provide
comprehensive numerical evidence that it does yield near-optimal results after
significantly shorter runtimes. For example, 10000 samples in $\mathbb{R}^2$
are processed in two seconds, rather than in $\approx 14$ hours required by the
previous approach to terminate. For higher dimensions, density estimation
becomes tractable as well: Processing $10000$ samples in $\mathbb{R}^6$
requires 35 minutes. The software is publicly available as CRAN R package
fmlogcondens.
