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Explore a mathematical lecture on transport-relaxed Schrödinger bridges that modifies classical marginal constraints by introducing penalty terms for transport costs between bridge marginals and prescribed marginals. Learn how this relaxation leads to a duality formula and reduces to finite-dimensional concave optimization when dealing with discrete prescribed marginals and absolutely continuous reference distributions. Discover the theoretical foundations including existence and uniqueness results for both primal and dual formulations, and understand the limiting behavior as penalty parameters approach infinity, revealing connections to discrete Schrödinger bridges with logarithmic blow-up characteristics. Examine two computational approaches - gradient ascent and Sinkhorn-type algorithms - both achieving linear convergence rates for numerical solutions. Gain insights into advanced probabilistic mass transport theory through this specialized treatment of relaxed optimal transport problems presented at the Erwin Schrödinger International Institute workshop on probabilistic mass transport and stochastic analysis.
