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We initiate a study of the following problem: Given a continuous domain$\Omega$ along with its convex hull $\mathcal{K}$, a point $A \in \mathcal{K}$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 inmathematics, quantum mechanics, statistics, and theoretical computer science.Our technical contributions include a polynomial bound on the norm of theoptimizer of the dual problem that holds in a very general setting and relieson a "balance" property of the measure $\mu$ on $\Omega$, and exact algorithmsfor evaluating the dual and its gradient for several interesting settings of$\Omega$ and $\mu$. Together, along with the ellipsoid method, these resultsimply polynomial-time algorithms to compute such KL-divergence minimizingdistributions in several cases. Applications of our results include: 1) anoptimization characterization of the Goemans-Williamson measure that is used toround a positive semidefinite matrix to a vector, 2) the computability of theentropic barrier for polytopes studied by Bubeck and Eldan, and 3) apolynomial-time algorithm to compute the barycentric quantum entropy of adensity matrix that was proposed as an alternative to von Neumann entropy inthe 1970s: this corresponds to the case when $\Omega$ is the set of rank oneprojections matrices and $\mu$ corresponds to the Haar measure on the unitsphere. Our techniques generalize to the setting of Hermitian rank $k$projections using the Harish-Chandra-Itzykson-Zuber formula, and are applicableeven beyond, to adjoint orbits of compact Lie groups.

On the computability of continuous maximum entropy distributions with applications