We study the following problem: Given a continuous domain Omega along with its convex hull K, a point A is an element of K, and a measure mu on Omega, find the probability density over Omega whose marginal is A and that minimizes the KL divergence to the uniform density with respect to mu. Several distributions in mathematics, physics, statistics, and theoretical computer science arise by different settings of the parameters of this problem. We give 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 [M. X. Goemans and D. P. Williamson, J ACM, 42 (1995), pp. 1115-1145] that is used to round a positive semidefinite matrix to a vector; (2) the computability of the entropic barrier for convex bodies, given a strong integration oracle, studied by [S. Bubeck and R. Eldan, Proc. Mach. Learn. Res. (PMLR), 40 (2015), p. 279], 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 [W. Band and J. L. Park, Found. Phys., 6 (1976), pp. 249-262; J. L. Park and W. Band, Found. Phys., 7 (1977), pp. 233-244; P. B. Slater, Phys. Lett. A, 159 (1991), pp. 411-414]; this corresponds to the case when Omega is the set of rank-one projection matrices and mu is derived from the Haar measure on the unit sphere.

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver's inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvari, which settled Friedland's lower matching conjecture.

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.