# A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities

arXiv: https://arxiv.org/abs/1812.05524 · HTML: https://mlsanity.com/paper/1812.05524/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2018.

## Built on by (most cited first, 1 of 1)

- [An Efficient Minimax Optimal Estimator For Multivariate Convex Regression](https://mlsanity.com/paper/2205.03368/) (2022)

## Abstract (arXiv)

We study the problem of computing the maximum likelihood estimator (MLE) of multivariate log-concave densities. Our main result is the first computationally efficient algorithm for this problem. In more detail, we give an algorithm that, on input a set of n points in ℝd and an accuracy parameter ε>0, it runs in time poly(n, d, 1/ε), and outputs a log-concave density that with high probability maximizes the log-likelihood up to an additive ε. Our approach relies on a natural convex optimization formulation of the underlying problem that can be efficiently solved by a projected stochastic subgradient method. The main challenge lies in showing that a stochastic subgradient of our objective function can be efficiently approximated. To achieve this, we rely on structural results on approximation of log-concave densities and leverage classical algorithmic tools on volume approximation of convex bodies and uniform sampling from convex sets.