Linear optimal transport embedding: provable Wasserstein classification for certain rigid transformations and perturbations

Abstract

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.

Supervised learning of sheared distributions using linearized optimal transport

“…This property describes an interplay between LOT and the pushforward operator, or in terms of Riemannian geometry, the invertability of the exponential map [14]. Similarly, small perturbations of the distributions in these classes can still be linearly separated under certain minimal separation conditions [20].…”

“…We use the optimal transport plan or map to build an embedding of probability measures into an L 2 -space known as "Linear Optimal Transportation" (LOT) [1,14,20,24,31] or "Monge embedding" [19]. LOT is a set of transformations based on optimal transport maps, which map a distribution μ to the optimal transport map that takes a fixed reference distribution σ to μ:…”

“…Indeed, it has been demonstrated in various applications that within the LOT embedding space, classes of probability measures can be well separated with linear machine learning tools. The main applications concern signal and image classification tasks [17,20,24], such as distinguishing facial expressions, separating healthy from cancerous tissue classes [30], and visualizing phenotypic differences between types of cells [6].…”

“…1 This has to do with the fact that in higher dimensions, there is a large family of potential group actions that can be applied to a distribution μ i (e.g., shifts, scalings, shearings, rotations), and μ σ contains a large number of measure preserving maps. It has been shown that shifts and scalings behave well with respect to the LOT embedding [1,20,24], meaning that two classes of probability measures obtained from scaling or shifting of a fixed measure can be linearly separated in the LOT embedding space. The reason lies in a property we refer to as the "compatibility condition", which is satisfied by shifts and scalings [1,20].…”

“…It has been shown that shifts and scalings behave well with respect to the LOT embedding [1,20,24], meaning that two classes of probability measures obtained from scaling or shifting of a fixed measure can be linearly separated in the LOT embedding space. The reason lies in a property we refer to as the "compatibility condition", which is satisfied by shifts and scalings [1,20]. This property describes an interplay between LOT and the pushforward operator, or in terms of Riemannian geometry, the invertability of the exponential map [14].…”