The Monge-Kantorovich Optimal Transport Distance for Image Comparison

TL;DR

This paper explores the use of the Monge-Kantorovich optimal transport distance, specifically the $L^2$ Wasserstein distance, for image comparison within machine learning, showing its effectiveness on MNIST data.

Contribution

It demonstrates the application of the $L^2$ Wasserstein distance in neural network architectures and compares its performance to other optimal transport formulations.

Findings

Wasserstein distance achieves excellent results on MNIST.

Comparison shows Wasserstein outperforms other distances in image recognition.

Optimal transport distances are effective for image comparison tasks.

Abstract

This paper focuses on the Monge-Kantorovich formulation of the optimal transport problem and the associated $L^2$ Wasserstein distance. We use the $L^2$ Wasserstein distance in the Nearest Neighbour (NN) machine learning architecture to demonstrate the potential power of the optimal transport distance for image comparison. We compare the Wasserstein distance to other established distances - including the partial differential equation (PDE) formulation of the optimal transport problem - and demonstrate that on the well known MNIST optical character recognition dataset, it achieves excellent results.