Optimal Transport for Machine Learners

A new set of course notes by mathematician Gabriel Peyré positions optimal transport as a unifying mathematical framework for modern machine learning, detailing how the theory of moving mass between probability distributions underpins generative models, domain adaptation, and network training dynamics. The work, titled "Optimal Transport for Machine Learners," was submitted to the arXiv preprint server on 10 May 2025 and revised on 14 Jun 2026 [1]. Optimal transport (OT) provides a method for comparing probability measures by determining the most efficient way to move mass from one distribution to another, combining a statistically meaningful measure of discrepancy with a geometry of interpolation [1]. The text is structured to guide readers from foundational concepts to computational techniques. It begins with finite assignment problems and the Monge map viewpoint, then progresses to Kantorovich couplings and dual potentials before explaining the algorithmic ideas that make transport practical for computation [1]. These algorithms include linear programming, semi-discrete solvers, and entropic regularization through Sinkhorn scaling [1][4]. Peyré's notes are explicitly positioned as an intermediate reference, bridging the computational focus of the existing book by Peyré and Cuturi and the theoretical emphasis of Santambrogio's work [3]. The material covers the Monge and Kantorovich formulations, Brenier's theorem, the dual and dynamic formulations, the Bures metric on Gaussian distributions, and gradient flows [2]. The final chapters address variants most relevant to contemporary machine learning, including divergences and adversarial losses, entropic and unbalanced relaxations, robust or spectral ground geometries, and Gromov and quantum extensions [1]. The notes detail how OT concepts are applied to analyze and design machine learning systems. Applications include training neural networks via gradient flows, modeling token dynamics in transformers, and understanding the structure of generative adversarial networks (GANs) and diffusion models [2][4]. The theory's ability to define a distance between probability measures, known as the Wasserstein distance, is highlighted as a key property for these applications [4]. The document focuses primarily on the mathematical content rather than deep learning implementation techniques [2]. The initial submission was 117 KB, with a revised version weighing 10,649 KB, reflecting the addition of substantial material [1].