Sliced Orlicz-Wasserstein
A paper proposes sliced Orlicz-Wasserstein distance, proving metric properties and cheaper estimation than Orlicz-Wasserstein.
The paper introduces sliced Orlicz-Wasserstein distance, replacing the Lp norm in sliced Wasserstein with a Luxemburg norm induced by an Orlicz function, and shows it is a metric that recovers sliced Wasserstein when the function is x^p. It proves that SOW convergence implies weak convergence, with a converse under compact support, and derives minimax-optimal sample complexity for the distance and a powered functional. Approximation uses Monte Carlo estimation and bisection search; experiments report better efficiency than Orlicz-Wasserstein and more flexibility than sliced Wasserstein on two-sample tests and generative-model evaluation.
- SOW generalizes sliced Wasserstein by replacing the Lp cost with an Orlicz Luxemburg norm.
- The paper proves metric, topology, sample-complexity, and minimax optimality results.
- Monte Carlo and bisection approximate SOW more cheaply than full Orlicz-Wasserstein.
Full article205 words · extracted from arxiv.org · click to collapse
We propose sliced Orlicz-Wasserstein (SOW) distance which is a generalization of sliced Wasserstein (SW) distance. SOW replaces the $L^p$ norm in SW with a Luxemburg norm cost induced by an Orlicz function $φ$. First, we prove that SOW distance is a metric on the space of measures with finite Orlicz norm, and show that it recovers the SW distance when the Orlicz function is $φ(x)=x^p$. Next, we derive the topological properties of the SOW distance. In particular, we show that convergence under SOW implies weak convergence, and the converse is true under the compact support condition. We then present the theoretical results for estimating the SOW distance. We derive sample complexity for both the distance itself and the powered functional of the distance, and prove their minimax optimality. In addition, we discuss the computational algorithm for approximating the SOW distance by Monte-Carlo estimation and bisection search, as well as the associated approximation error and computational complexity analysis. Our experimental results reveal the superior computational efficiency of SOW compared with Orlicz-Wasserstein (OW) distance. Also, in the experiments, we demonstrate the favorable flexibility of SOW distance over SW in detecting differences between distributions by comparing their performance in two-sample tests and evaluating generative models on image datasets.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.32847