# A new computational framework for log-concave density estimation

arXiv: https://arxiv.org/abs/2105.11387 · HTML: https://mlsanity.com/paper/2105.11387/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2021.

## Built on by (most cited first, 4 of 4)

- [Universal Inference Meets Random Projections: A Scalable Test for Log-concavity](https://mlsanity.com/paper/2111.09254/) (2021)
- [Nonparametric inference under shape constraints: past, present and future](https://mlsanity.com/paper/2509.26040/) (2025)
- [An Efficient Minimax Optimal Estimator For Multivariate Convex Regression](https://mlsanity.com/paper/2205.03368/) (2022)
- [Log-concave density estimation in undirected graphical models](https://mlsanity.com/paper/2206.05227/) (2022)

## Abstract (arXiv)

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 textttLogConcComp (Log-Concave Computation).