Abstract:
We initiate a study of the following problem: Given a continuous domain Ω along with its convex hull K, a point A ∈ 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 by Band and Park in the 1970s: this corresponds to the case when Ω is the set of rank one projection matrices and µ 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.
