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

The log-concave maximum likelihood estimator (MLE) problem answers: for a set of points X_1,...X_n ∈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, 1/ϵ,r) to compute a log-concave distribution whose log-likelihood is at most ϵ less than that of the MLE, and r is parameter of the problem that is bounded by the ℓ_2 norm of the vector of log-likelihoods the MLE evaluated at X_1,...,X_n.