A Polynomial Time Algorithm for Maximum Likelihood Estimation of
Multivariate Log-concave Densities
(2018)cite arxiv:1812.05524.Abstract
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 $R^d$ and an
accuracy parameter $\epsilon>0$, it runs in time $poly(n, d,
1/\epsilon)$, and outputs a log-concave density that with high probability
maximizes the log-likelihood up to an additive $\epsilon$. 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.
Description
[1812.05524] A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities
