The paper introduces a new family of optimal transportation distances called Sinkhorn distances, which are computed using the Sinkhorn-Knopp matrix scaling algorithm. These distances are derived from a maximum-entropy perspective by regularizing the classical optimal transportation problem with an entropic term. The resulting optimization problem is strictly convex and can be solved efficiently, offering significant speed improvements over traditional methods. The Sinkhorn distances are shown to perform better than classical optimal transportation distances on the MNIST benchmark problem and are several orders of magnitude faster, making them suitable for large-scale data analysis. The paper also discusses the metric properties of Sinkhorn distances and provides empirical results demonstrating their effectiveness and computational efficiency.The paper introduces a new family of optimal transportation distances called Sinkhorn distances, which are computed using the Sinkhorn-Knopp matrix scaling algorithm. These distances are derived from a maximum-entropy perspective by regularizing the classical optimal transportation problem with an entropic term. The resulting optimization problem is strictly convex and can be solved efficiently, offering significant speed improvements over traditional methods. The Sinkhorn distances are shown to perform better than classical optimal transportation distances on the MNIST benchmark problem and are several orders of magnitude faster, making them suitable for large-scale data analysis. The paper also discusses the metric properties of Sinkhorn distances and provides empirical results demonstrating their effectiveness and computational efficiency.