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We consider the problem of computing the maximum likelihood multivariatelog-concave distribution for a set of points. Specifically, we present analgorithm which, given $n$ points in $\mathbb{R}^d$ and an accuracy parameter$\epsilon>0$, runs in time $poly(n,d,1/\epsilon),$ and returns a log-concavedistribution which, with high probability, has the property that the likelihoodof the $n$ points under the returned distribution is at most an additive$\epsilon$ less than the maximum likelihood that could be achieved via anylog-concave distribution. This is the first computationally efficient(polynomial time) algorithm for this fundamental and practically importanttask. Our algorithm rests on a novel connection with exponential families: themaximum likelihood log-concave distribution belongs to a class of structureddistributions which, while not an exponential family, "locally" possesses keyproperties of exponential families. This connection then allows the problem ofcomputing the log-concave maximum likelihood distribution to be formulated as aconvex optimization problem, and solved via an approximate first-order method.Efficiently approximating the (sub) gradients of the objective function of thisoptimization problem is quite delicate, and is the main technical challenge inthis work.

A Polynomial Time Algorithm for Log-Concave Maximum Likelihood via Locally Exponential Families