A new computational framework for log-concave density estimation

TL;DR

This paper introduces a novel computational framework for log-concave density estimation that improves algorithm efficiency and convergence, facilitating practical application in statistical inference.

Contribution

The paper proposes new algorithms based on smoothing techniques that enhance the computational efficiency and convergence properties of log-concave density estimation.

Findings

Algorithms achieve $1/T$ convergence rate with high probability

Framework demonstrated on synthetic and real datasets

Implementation available in open-source repository

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 \texttt{LogConcComp} (Log-Concave Computation).