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