On the computability of continuous maximum entropy distributions with applications

We initiate a study of the following problem: Given a continuous domain $Ω$ along with its convex hull $\mathcal{K}$, a point $A \in \mathcal{K}$ and a prior measure $μ$ on $Ω$, find the probability density over $Ω$ whose marginal is $A$ and that minimizes the KL-divergence to $μ$. 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 $μ$ on $Ω$, and exact algorithms for evaluating the dual and its gradient for several interesting settings of $Ω$ and $μ$. 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 in the 1970s: this corresponds to the case when $Ω$ is the set of rank one projections matrices and $μ$ corresponds to the Haar measure on the unit sphere. Our techniques generalize to the setting of Hermitian rank $k$ projections using the Harish-Chandra-Itzykson-Zuber formula, and are applicable even beyond, to adjoint orbits of compact Lie groups.