Uniqueness of the Gaussian KL-UOT barycenter at intermediate penalties

Determine whether the Gaussian-restricted KL-unbalanced optimal transport barycenter is unique for intermediate penalty values, where neither the large-penalty Wasserstein limit nor the small-penalty Chernoff-affinity limit directly governs the problem.

Background

The paper establishes uniqueness of the Gaussian-restricted barycenter for sufficiently large common penalty scales and characterizes the small-penalty limit through a possibly non-singleton maximizer set of a weighted Chernoff-affinity functional. For general intermediate penalty values, the profiled objective admits global minimizers and the reverse-KL MM iteration converges from every nondegenerate initialization to a stationary fixed point, but these results do not establish that the stationary point or global minimizer is unique.

Numerical experiments report no distinct attracting stationary point for the tested configurations, but the authors explicitly distinguish this empirical observation from a proof of global uniqueness. Establishing or disproving uniqueness in the intermediate regime would clarify the transition between displacement-based averaging and affinity-based overlap maximization.