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arXiv · 1811.03204

An Efficient Algorithm for High-Dimensional Log-Concave Maximum Likelihood

Brian Axelrod·Gregory Valiant
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Abstract

The log-concave maximum likelihood estimator (MLE) problem answers: for a set of points $X_1,...X_n \in \mathbb R^d$, which log-concave density maximizes their likelihood? We present a characterization of the log-concave MLE that leads to an algorithm with runtime $poly(n,d, \frac 1 \epsilon,r)$ to compute a log-concave distribution whose log-likelihood is at most $\epsilon$ less than that of the MLE, and $r$ is parameter of the problem that is bounded by the $\ell_2$ norm of the vector of log-likelihoods the MLE evaluated at $X_1,...,X_n$.

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Brian Axelrod, Gregory Valiant. 2018-11-08. An Efficient Algorithm for High-Dimensional Log-Concave Maximum Likelihood. https://arxiv.org/abs/1811.03204

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