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A new computational framework for log-concave density estimation

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Chen, Wenyu 
Mazumder, Rahul 
Samworth, Richard J 


jats:titleAbstract</jats:title>jats:pIn statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order 1/jats:italicT</jats:italic> up to logarithmic factors on the objective function scale, where jats:italicT</jats:italic> denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository (Log-Concave Computation).</jats:p>



math.OC, stat.CO, stat.CO, stat.ME

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Mathematical Programming Computation

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Springer Science and Business Media LLC
European Commission Horizon 2020 (H2020) ERC (101019498)
Engineering and Physical Sciences Research Council (EP/P031447/1)
Engineering and Physical Sciences Research Council (EP/N031938/1)