Emergent Mind
An Efficient Algorithm for High-Dimensional Log-Concave Maximum Likelihood
(1811.03204)
Published Nov 8, 2018
in
cs.DS
and
stat.CO
Abstract
The log-concave maximum likelihood estimator (MLE) problem answers: for a set of points $X1,...Xn \in \mathbb Rd$, 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 $\ell2$ norm of the vector of log-likelihoods the MLE evaluated at $X1,...,X_n$.
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