Collaborative Research: Generalized Slicers for Efficient and Versatile Computation of Transport-Based Dissimilarities and Assignments

Modern artificial intelligence and machine learning systems often learn by comparing complex data: determining when two collections of measurements are similar, how they differ, and how they can be aligned. Optimal Transport provides a rigorous framework for these comparisons by modeling data as mass distributions and measuring the minimal cost of transforming one distribution into another. Because this transformation tracks how mass is moved, reshaped, and reallocated, Optimal Transport provides both a measure of difference and a meaningful alignment between datasets. Despite its growing importance, current algorithms are often too computationally expensive for modern datasets, which may contain millions of measurements or represent complex objects such as text, images, videos, or sensor readings. This project develops new mathematical and computational tools to accelerate transport-based comparisons. The approach centers on geometry-aware ways of simplifying complex data by breaking it into simpler but still informative views. These tools, called Generalized Slicers, are designed to make Optimal Transport more scalable, enabling larger datasets to be analyzed at lower computational cost, and more interpretable, helping researchers understand meaningful relationships rather than only a numerical score. By creating mathematically grounded tools for reliable data analysis and alignment, the work advances data-driven scientific discovery. The outcomes are expected to improve methods for studying living systems, support safer engineered systems, and train the next generation of mathematical and computational scientists. Potential benefits include helping scientists compare biological data across developmental stages, helping engineers detect damage in vehicles and infrastructure from sensor readings, providing open-source software, mentoring students, creating new course and minicourse materials, and broadening participation in mathematical and computational sciences.     The goal of this project is to develop a unified mathematical, statistical, and computational framework for studying high-dimensional measures supported on metric spaces through Generalized Slicers: flexible operators that generalize classical linear slicing methods in Optimal Transport. These generalized slices are lower-dimensional measures obtained by integrating the original measure along families of hypersurfaces, thereby providing tractable statistical summaries of complex measures while preserving geometric information. The project studies two complementary strategies. The first uses iterative alignment methods that progressively match measures by optimizing along individual generalized slices. The second uses expectation-based approaches that compute many lower-dimensional transport assignments and combine them in the original data space. Across these two directions, the project investigates slicers that may be linear or nonlinear, deterministic or stochastic, and fixed or learnable; establishes convergence guarantees, sample-complexity results, and stability theory; and extends the theory to measures supported on manifolds and to unbalanced settings where data may be noisy or incomplete and total mass may change. The research also develops transport-based assignments and interpolation methods that preserve geometric structure while remaining computationally efficient for high-dimensional data. As an additional outcome, the project explores inverse problems for positive definite operators, with potential impact in matrix-free and streaming settings. The methods are validated in two application areas: computational biology, with an emphasis on robust cross-stage alignment and reconstruction of spatiotemporal gene-expression trajectories in transcriptomics; and engineering, with an emphasis on structural health monitoring, damage detection, and temporal visualization of infrastructure conditions. The project produces mathematical theory, scalable algori NSF Award ID: 2603774 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Soheil Kolouri | Institution: Vanderbilt University, NASHVILLE, TN | Award Amount: $149,835 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2603774 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2603774.html