Monge's Optimal Transport Distance for Image Classification

Abstract: This paper focuses on a similarity measure, known as the Wasserstein distance, with which to compare images. The Wasserstein distance results from a partial differential equation (PDE) formulation of Monge's optimal transport problem. We present an efficient numerical solution method for solving Monge's problem. To demonstrate the measure's discriminatory power when comparing images, we use a $1$-Nearest Neighbour ($1$-NN) machine learning algorithm to illustrate the measure's potential benefits over other more traditional distance metrics and also the Tangent Space distance, designed to perform excellently on the well-known MNIST dataset. To our knowledge, the PDE formulation of the Wasserstein metric has not been presented for dealing with image comparison, nor has the Wasserstein distance been used within the $1$-nearest neighbour architecture.