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 , r) to compute a log-concave distribution whose log-likelihood is at most less than that of the MLE, and r is parameter of the problem that is bounded by the 2 norm of the vector of log-likelihoods the MLE evaluated at X 1 , ..., X n .
Contemporaneous work by Axelrod and Valiant REF gives a poly(n, d, r, 1/ǫ) time algorithm to compute an ǫ-approximation log-concave MLE, where r is a parameter bounded by the ℓ 2 -norm of the log-likelihoods of the input points x 1 , . . . , x n under f n .
An Efficient Algorithm for High-Dimensional Log-Concave Maximum Likelihood
ArXiv
ArXiv
Mathematics Computer Science