Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations
This provides a computationally efficient method for distribution classification, particularly useful in scientific fields dealing with rigid transformations and perturbations, though it is incremental as it builds on existing LOT concepts.
The paper tackles the problem of discriminating between distributions by introducing Linear Optimal Transport (LOT) embedding, which embeds distributions into an L^2-space for linear separability and proves conditions where it is nearly isometric to Wasserstein-2 distance, reducing computational cost from N^2 to N optimal transport maps for N distributions.
Discriminating between distributions is an important problem in a number of scientific fields. This motivated the introduction of Linear Optimal Transportation (LOT), which embeds the space of distributions into an $L^2$-space. The transform is defined by computing the optimal transport of each distribution to a fixed reference distribution, and has a number of benefits when it comes to speed of computation and to determining classification boundaries. In this paper, we characterize a number of settings in which LOT embeds families of distributions into a space in which they are linearly separable. This is true in arbitrary dimension, and for families of distributions generated through perturbations of shifts and scalings of a fixed distribution.We also prove conditions under which the $L^2$ distance of the LOT embedding between two distributions in arbitrary dimension is nearly isometric to Wasserstein-2 distance between those distributions. This is of significant computational benefit, as one must only compute $N$ optimal transport maps to define the $N^2$ pairwise distances between $N$ distributions. We demonstrate the benefits of LOT on a number of distribution classification problems.