Applied Mathematics Colloquium – Shiying Li, University of North Carolina at Chapel Hill

Title: Slice-Matching for Measure Transport: Approximation and Iterative Schemes

Speaker: Shiying Li, Department of Mathematics, University of North Carolina at Chapel Hill

Abstract: Transporting and estimating probability measures are fundamental tasks in various generative modeling methods like normalizing flows. An equally crucial aspect is having a suitable metric to gauge model performance and guide algorithm design, particularly for scalability. Our focus lies on an iterative slice-matching approach initially introduced by Pitie et al. for transferring color statistics. We establish a connection with the gradient descent perspective of the sliced-Wasserstein distance, which offers computationally efficient closed-form solutions, especially beneficial in high-dimensional scenarios. While these approaches have proven effective in data science applications, their convergence characteristics have received limited research attention up to now.

The talk will be structured into two main segments: the first part centers on an almost sure convergence proof for a broader formulation of such schemes, while the second delves into the expressive capabilities of a single step within these iterative methods. We will delve into properties such as invariance, equivariance, and Lipschitz properties associated with the slice-matching operator, which yield recovery and stability outcomes for these approximations. If time permits, we will also explore various associated affine registration problems and their relationship with slice-matching’s ability to incorporate transformations like shifts and scaling in the initial step. The talk is based on joint work with Caroline Moosmueller.

Biosketch: Shiying Li received her Ph.D. in Mathematics in 2019 from Vanderbilt University. Currently she holds the position of Postdoctoral Research Associate in the Department of Mathematics at the University of North Carolina at Chapel Hill. Prior to her current position, she was a postdoc at the Department of Biomedical Engineering at the University of Virginia where she worked on projects centered around transport and other Lagrangian transforms with applications in image and signal analysis. Her current work primarily focuses on applied optimal transport and mathematical data science.