Embedding Empirical Distributions for Computing Optimal Transport Maps

This work was performed while the first author was interning at Hong Kong Baptist University. Abstract Distributional data have become increasingly prominent in modern signal processing, highlighting the necessity of computing optimal transport (OT) maps across multiple probability distributions. Nevertheless, recent studies on neural OT methods predominantly focused on the efficient computation of a single map between two distributions. To address this challenge, we introduce a novel approach to learning transport maps for new empirical distributions. Specifically, we employ the transformer architecture to produce embeddings from distributional data of varying length; these embeddings are then fed into a hypernetwork to generate neural OT maps. V arious numerical experiments were conducted to validate the embeddings and the generated OT maps. Optimal transport (OT) theory [1] is a mathematical framework for finding the most efficient way (in the sense of minimizing a given cost function) to transport one probability distribution to another. When the quadratic cost is used, OT theory induces a metric space for probability measures, and the distance thereof is referred to as the 2-Wasserstein metric [2]. This notion provides a geometric view of distributions, and therefore makes OT an invaluable tool in information theory [3]-[6]. Furthermore, OT has already been used in many applications, such as flow-based diffusion models [7], [8], GANs [9], [10], style transfer [11], data embedding [12], [13], multilingual alignment [14], [15], domain adaptation [16], [17], and model compression [18]-[20].