Fast Multivariate Log-Concave Density Estimation

Abstract

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.