Unbalanced optimal transport (UOT) has recently gained much attention due to its flexible framework for handling un-normalized measures and its robustness properties. In this work, we explore learning (structured) sparse transport plans in the UOT setting, i.e., transport plans have an upper bound on the number of non-sparse entries in each column (structured sparse pattern) or in the whole plan (general sparse pattern). We propose novel sparsity-constrained UOT formulations building on the recently explored maximum mean discrepancy based UOT. We show that the proposed optimization problem is equivalent to the maximization of a weakly submodular function over a uniform matroid or a partition matroid. We develop efficient gradient-based discrete greedy algorithms and provide the corresponding theoretical guarantees. Empirically, we observe that our proposed greedy algorithms select a diverse support set and we illustrate the efficacy of the proposed approach in various applications.

翻译：非平衡最优传输（UOT）因其处理非归一化度量的灵活框架及其鲁棒性特性，近年来受到广泛关注。本文探索在UOT框架下学习（结构化）稀疏传输方案，即传输方案中每列（结构化稀疏模式）或整个方案（通用稀疏模式）的非稀疏条目数量存在上界。基于近期探索的基于最大均值差异的UOT方法，我们提出了新颖的稀疏约束UOT模型。我们证明所提出的优化问题等价于在均匀拟阵或划分拟阵上对弱次模函数进行最大化。我们开发了高效的基于梯度的离散贪心算法，并提供了相应的理论保证。实证结果表明，我们提出的贪心算法能够选择多样化的支撑集，并通过多种应用场景验证了所提方法的有效性。