Hybrid Wasserstein Distance: An Approximation for Optimal Transport Distances

Projection-based variants of optimal transport, such as the Sliced Wasserstein (SW) and its extensions, have become popular alternatives to classical Wasserstein distances due to their scalability and analytical tractability. However, most of these methods rely on independently sampled random projections, which often fail to capture semantically meaningful directions, leading to inefficiencies and limited expressiveness, especially in high-dimensional settings. In this work, we propose the Hybrid Merging Projection Wasserstein (HW) distance, a novel and efficient alternative that addresses these limitations by combining data-driven and random projections in a principled way. At the core of HW is the Linear Merging Projection (LMP), a new projection technique designed to minimize between-class variance, thereby promoting smooth alignment between distributions. HW incorporates random directions as well to achieve a balance between structural awareness and projection diversity. We evaluate HW across a range of synthetic and real-world benchmarks, including color transfer and distribution alignment tasks, to demonstrate the favorable performance of the proposed HW.