发表机构
Fudan University（复旦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带独立边际惩罚的二次双侧KL不平衡最优传输，研究高斯限制重心的变分理论，构造反向KL MM迭代并通过数值实验验证其收敛性质。
AI 中文摘要
我们研究带独立边际惩罚且无耦合熵的二次双侧Kullback-Leibler不平衡最优传输的高斯限制重心。重心质量的精确剖析将问题简化为具有内生Gibbs权重的平滑高斯形状泛函。我们建立全局可达性，推导平稳矩方程，并构造反向KL majorization-minimization（MM）迭代，该迭代的完整序列从任意非退化高斯初始化收敛至平稳不动点。对角二阶变分诱导耦合Bures-Wasserstein与Fisher-Rao度量的平行和张量；其有限质量扩展容许径向锥表示。在常见惩罚缩放下，全局极小元收敛至带有效权重的高斯Wasserstein重心；对于足够大的惩罚，极小元唯一且容许一阶解析展开。在小惩罚 regime下，每个全局极小元到加权Chernoff亲和泛函的紧极大元集的距离关于均值-协方差参数距离消失。数值实验展示MM下降、局部收缩及两种惩罚极限。
英文摘要
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.
