Fast Multivariate Log-Concave Density Estimation
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We present a computational approach to log-concave density estimation. The state-of- the-art approach of Cule et al. (2010) utilizes 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 ≥ 2 and large sample sets. Our approach is based on mildly non-convex smooth approximations of the objective function and sparse, adaptive piecewise-affine density parametrization. Established memory-efficient numerical optimization techniques enable to process larger data sets for dimensions d ≥ 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, after significantly shorter runtimes. For example, a sample set of size n = 10000 in R^2 is processed in less then a second, rather than in ≈ 5 hours required by the approach of (Cule et al., 2010) to terminate. For higher dimensions, density estimation becomes tractable as well: Processing a sample set of size n = 10000 in R^6 requires 35 minutes. The software is publicly available as CRAN R package fmlogcondens.
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