On the Computability of Continuous Maximum Entropy Distributions with Applications

We initiate a study of the following problem: Given a continuous domain Omega along with its convex hull K, a point A is an element of K and a prior measure mu on Omega, find the probability density over Omega whose marginal is A and that minimizes the KL-divergence to mu. This framework gives rise to several extremal distributions that arise in mathematics, quantum mechanics, statistics, and theoretical computer science. Our technical contributions include a polynomial bound on the norm of the optimizer of the dual problem that holds in a very general setting and relies on a "balance" property of the measure mu on Omega, and exact algorithms for evaluating the dual and its gradient for several interesting settings of Omega and mu. Together, along with the ellipsoid method, these results imply polynomial-time algorithms to compute such KL-divergence minimizing distributions in several cases. Applications of our results include: 1) an optimization characterization of the Goemans-Williamson measure that is used to round a positive semidefinite matrix to a vector, 2) the computability of the entropic barrier for polytopes studied by Bubeck and Eldan, and 3) a polynomial-time algorithm to compute the barycentric quantum entropy of a density matrix that was proposed as an alternative to von Neumann entropy by Band and Park in the 1970s: this corresponds to the case when Omega is the set of rank one projection matrices and mu corresponds to the Haar measure on the unit sphere. Our techniques generalize to the setting of rank k projections using the Harish-Chandra-Itzykson-Zuber formula, and are applicable even beyond, to adjoint orbits of compact Lie groups.