Gaussian-Restricted Barycenters for KL-Unbalanced Optimal Transport: Variational Theory and Fixed-Point Convergence
We study Gaussian-restricted barycenters for quadratic two-sided Kullback--Leibler unbalanced optimal transport with independent marginal penalties and no coupling entropy. Exact profiling of the barycenter mass reduces the problem to a smooth Gaussian shape functional with endogenous Gibbs weights. We establish global attainment, derive the stationary moment equations, and construct a reverse-KL majorization--minimization (MM) iteration whose full sequence converges from every nondegenerate Gaussian initialization to a stationary fixed point. The diagonal second variation induces a parallel-sum tensor coupling the Bures--Wasserstein and Fisher--Rao metrics; its finite-mass extension admits a radial cone representation. Under common penalty scaling, global minimizers converge to a Gaussian Wasserstein barycenter with effective weights; for sufficiently large penalties, the minimizer is unique and admits a first-order analytic expansion. In the small-penalty regime, the distance of every global minimizer to the compact maximizer set of a weighted Chernoff affinity functional vanishes with respect to the mean--covariance parameter distance. Numerical experiments illustrate MM descent, local contraction, and the two penalty limits.