- Injective Sliced-Wasserstein embedding for weighted units and level clouds
Summary: We current the Sliced Wasserstein Embedding — a novel methodology to embed multisets and distributions over Rd into Euclidean area. Our embedding is injective and roughly preserves the Sliced Wasserstein distance. Furthermore, when restricted to multisets, it’s bi-Lipschitz. We additionally show that it’s unimaginable to embed distributions over Rd right into a Euclidean area in a bi-Lipschitz method, even below the idea that their help is bounded and finite. We show empirically that our embedding affords sensible benefit in studying duties over present strategies for dealing with multisets.
2. Statistical and Computational Ensures of Kernel Max-Sliced Wasserstein Distances
Authors: Jie Wang, March Boedihardjo, Yao Xie
Summary: Optimum transport has been very profitable for varied machine studying duties; nevertheless, it’s recognized to undergo from the curse of dimensionality. Therefore, dimensionality discount is fascinating when utilized to high-dimensional information with low-dimensional constructions. The kernel max-sliced (KMS) Wasserstein distance is developed for this function by discovering an optimum nonlinear mapping that reduces information into 1 dimensions earlier than computing the Wasserstein distance. Nevertheless, its theoretical properties haven’t but been totally developed. On this paper, we offer sharp finite-sample ensures below milder technical assumptions in contrast with state-of-the-art for the KMS p-Wasserstein distance between two empirical distributions with n samples for basic p∈[1,∞). Algorithm-wise, we show that computing the KMS 2-Wasserstein distance is NP-hard, and then we further propose a semidefinite relaxation (SDR) formulation (which can be solved efficiently in polynomial time) and provide a relaxation gap for the SDP solution. We provide numerical examples to demonstrate the good performance of our scheme for high-dimensional two-sample testing.
