Optimal Transport Seminar

Organizers: Chris Gartland & Kevin McGoff

Overview: This Fall 2025 seminar will cover a broad range of topics related to optimal transport, determined by the interests of the participants. There will be an emphasis on expository talks, and graduate students are encouraged to attend and to give talks. We will typically meet weekly on Fridays at 3:00pm, location Fretwell 302. References for external reading are provided.

External Reading:

Upcoming Talk: Talks have concluded for the semester

Full Schedule:

  • Friday Nov 14, 2025
  • Time: 3:00pm
  • Location: Fret 302
  • Speaker: Helen Li
  • Title: A Quick Introduction to Diffusion Models, Schrödinger Bridges, and Entropy-Regularized Optimal Transport
  • Abstract: In this talk, I will begin with a tutorial overview of diffusion models and their applications in generative AI. I will then introduce a particular class of diffusion models known as diffusion bridges, focusing on the details of a special case—the Schrödinger bridge—and its connection to entropy-regularized optimal transport (EOT). Finally, I will discuss results from the paper “Plug-in Estimation of Schrödinger Bridges (https://arxiv.org/pdf/2408.11686),” which proposes a novel non-iterative method that yields a natural plug-in estimator of the time-dependent drift defining the bridge between the source and target measures.
  • Friday Nov 7, 2025
  • Time: 3:00pm
  • Location: Fret 302
  • Speaker: Yang Xiang (UNC Chapel Hill)
  • Title: Graph Joining and Disjointness
  • Abstract: We investigate a phenomenon called disjointness within the framework of graph joinings as a means to characterize structural incompatibility between graphs. Given two weighted, undirected graphs (equivalently, reversible Markov chains), we say they are disjoint if their only possible joining—that is, their only reversible Markovian coupling—is the product. This perspective shifts attention from computing distances between graphs to examining the set of their joinings, thereby highlighting structural features that underlie their incompatibility.

  • Friday Oct 31, 2025
  • Time: 3:00pm
  • Location: Fret 302
  • Speaker: Robert Murray-Gramlich
  • Title: Statistical Aspects of Wasserstein Distances
  • Abstract: Wasserstein distances have recently become popularized in statistics due to their geometric properties (and the topology it induces) as well as the many applications to machine and statistical learning. We will discuss the “Wasserstein Law of Large Numbers” which gives an upper bound on the convergence speed of W_p on a compact set and give a complete proof including the bound on the logarithmic factor in the “critical case”. Given time we will also discuss some open related problems.
  • Friday Oct 24, 2025
  • Time: 3:00pm
  • Location: Fret 302
  • Speaker: Phuong Hoang
  • Title: Optimal Graph Joining with Applications to Isomorphism Detection and Identification
  • Abstract: We introduce and develop a new constrained optimal transport problem for graphs, called the optimal graph joining (OGJ) problem, and study its relation to graph isomorphism. The graphs of interest are finite, undirected, and may be weighted and labeled. Extending the idea of probabilistic couplings to the setting of graphs, we first introduce the notion of a graph joining of two graphs G and H, which is a graph K on the product of the vertex sets of G and H that has G and H as marginals in an appropriate sense. Given two graphs and a vertex-based cost function, OGJ aims to find a graph joining that minimizes the expected cost. After establishing the basic properties of the OGJ problem, we provide theoretical results connecting the OGJ problem to the graph isomorphism problem. In particular, we provide a variety of sufficient conditions on graph families under which OGJ detects and identifies isomorphisms between graphs within the family.
  • Friday Oct 17, 2025
  • Time: 3:00pm
  • Location: Fret 302
  • Speaker: Helen Li
  • Title: Mean-Field Asymptotics of Entropic OT Hessian Spectra at Fixed Regularization on Random Point Clouds
  • Abstract: We investigate how the eigenvalues of a canonical Hessian matrix arising in entropic optimal transport (OT) behave as the sample size increases. Given N uniformly sampled points from a bounded region with Lipschitz boundary and a fixed regularization parameter \epsilon>0, we show that the smallest nonzero eigenvalue of the associated Sinkhorn matrix scales as c_\epsilon /N with high probability. The constant c_\epsilon is determined by the spectral gap of a Gaussian smoothing operator defined over \omega. In the limit of small \epsilon, the Gaussian smoothing operator approaches the Neumann Laplacian, revealing a geometric connection between OT regularization and diffusion. These findings provide a theoretical foundation for understanding the scaling and stability of large-scale entropic OT problems.
  • Friday Oct 3, 2025
  • Time: 3:00pm
  • Location: Fret 302
  • Speaker: Chris Gartland
  • Title: Introduction to Wasserstein Metrics II
  • Abstract: I will continue with the introduction to Wasserstein-1 metrics and transportation cost norms on graphs and metric spaces.
  • Friday Sept 26, 2025
  • Time: 3:30pm
  • Location: Fret 302
  • Speaker: Kevin McGoff
  • Title: Optimal transport in ergodic theory and dynamical systems
  • Abstract: In this expository talk, I will provide an introduction to the use of optimal transport in the setting of ergodic theory and dynamical systems. In this setting it is natural to consider a constrained set of couplings that respect the dynamics, called joinings. Originally introduced by Furstenberg in 1967, joinings have proved to be an influential and powerful tool for studying stationary dynamics. I will describe the setting, define joinings, and describe some of the ways that optimal transport has made an impact. No prior knowledge of dynamical systems will be assumed.
  • Friday Sept 19, 2025
  • Time: 3:00pm
  • Location: Fret 302
  • Speaker: Helen Li
  • Title: Condition number of Hessian of Entropy-Regularized OT and other types of modified optimal transport problem.
  • Abstract: I will begin by discussing the condition number for the benchmark example of equally spaced points on the unit circle, and then pose questions related to datasets consisting of random samples. I will also explain the main ideas and results from  https://arxiv.org/abs/2107.12364. Next, I will turn to other types of modified optimal transport with improved regularity properties, focusing in particular on unbalanced OT. For this part, I will primarily follow the results presented in https://arxiv.org/pdf/2211.08775 
  • Friday Sept 12, 2025
  • Time: 2:30pm
  • Location: Fret 302
  • Speaker: Helen Li
  • Title: Robust Numerical Differentiation for Entropy-regularized Optimal Transport (EOT) with application to Shuffled Regression
  • Abstract: In this presentation, I will begin by introducing shuffled regression and entropic optimal transport (EOT) as one possible approach. I will then discuss the derivatives of EOT, provide a brief overview of numerical condition numbers, and explain how to compute them robustly. I will present an example of shuffled regression that could serve as a potential benchmark for future numerical algorithm comparisons. Finally, I would like to discuss future work, extensions, and possible collaborations among the audience.
  • Friday Sept 5, 2025
  • Time: 2:30pm
  • Location: meet at Fret 302, but may have to move if room is taken
  • Speaker: Chris Gartland
  • Title: Introduction to Wasserstein Metrics
  • Abstract: I will give an introductory talk defining Wasserstein-p metrics, mostly focused on p=1. Basic properties and interesting questions will be discussed.