TY  - JOUR
U1  - Wissenschaftlicher Artikel
A1  - Chen, Wenyu
A1  - Mazumder, Rahul
A1  - Samworth, Richard J.
T1  - A new computational framework for log-concave density estimation
JF  - Mathematical Programming Computation
N2  - AbstractIn 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 (Log-Concave Computation).
AB  - AbstractIn 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 (Log-Concave Computation).
Y1  - 2024
SN  - 1867-2949
SS  - 1867-2949
U6  - https://doi.org/10.1007/s12532-024-00252-0
DO  - https://doi.org/10.1007/s12532-024-00252-0
VL  - 16
IS  - 2
SP  - 185
EP  - 228
S1  - 44
PB  - Springer Science and Business Media LLC
ER  -