A new computational framework for log-concave density estimation
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12532_2024_Article_252.pdf
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817efcff913d473f723bf9a093c74226
Author(s)
Chen, Wenyu
Mazumder, Rahul
Samworth, Richard J.
Date Issued
April 30, 2024
Journal
Mathematical Programming Computation
Publisher
Springer Science and Business Media LLC
Citation
Chen, W., Mazumder, R. & Samworth, R.J. A new computational framework for log-concave density estimation. Math. Prog. Comp. (2024).
Version
Final published version
Abstract
In 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/T up to logarithmic factors on the objective function scale, where T 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 LogConcComp (Log-Concave Computation).
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Sloan School of Management
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DOI of Published Version
https://doi.org/10.1007/s12532-024-00252-0
