Optimal transport (OT) theory has attracted much attention in machine learning and signal processing applications. OT defines a notion of distance between probability distributions of source and target data points. A crucial factor that influences OT-based distances is the ground metric of the embedding space in which the source and target data points lie. In this work, we propose to learn a suitable latent ground metric parameterized by a symmetric positive definite matrix. We use the rich Riemannian geometry of symmetric positive definite matrices to jointly learn the OT distance along with the ground metric. Empirical results illustrate the efficacy of the learned metric in OT-based domain adaptation.

翻译：最优传输理论在机器学习和信号处理应用中引起了广泛关注。最优传输定义了源数据点与目标数据点概率分布之间的一种距离度量。影响最优传输距离的关键因素在于源数据与目标数据所处的嵌入空间中的地面度量。本研究提出学习一种由对称正定矩阵参数化的潜在地面度量。我们利用对称正定矩阵丰富的黎曼几何结构，联合学习最优传输距离与地面度量。实证结果表明，学习得到的度量在基于最优传输的领域自适应任务中具有显著效果。