One for all and all for one: Efficient computation of partial Wasserstein distances on the line

Partial Wasserstein helps overcoming some of the limitations of Optimal Transport when the distributions at stake differ in mass, contain noise or outliers or exhibit mass mismatches across distribution modes.We introduce PAWL, a novel algorithm designed to efficiently compute exact PArtial Wasserstein distances on the Line. PAWL not only solves the partial transportation problem for a specified amount of mass to be transported, but _for all_ admissible mass amounts. This flexibility is valuable for machine learning tasks where the level of noise is uncertain and needs to be determined through cross-validation, for example. By achieving $O(n \log n)$ time complexity for the partial 1-Wasserstein problem on the line, it enables practical applications with large scale datasets. Additionally, we introduce a novel slicing strategy tailored to Partial Wasserstein, which does not permit transporting mass between outliers or noisy data points. We demonstrate the advantages of PAWL in terms of computational efficiency and performance in downstream tasks, outperforming existing (sliced) Partial Optimal Transport techniques.