Joint Stochastic Localization: Theory, Computation, and Applications

Modern scientific research often involves data in which each sample consists of many measurements, such as those arising in medical imaging, DNA sequencing, and physical experiments. Such data are best represented as probability distributions to capture information about the population as a whole. Analyzing such data helps reveal population-level mechanisms and is critical to many data-driven discoveries. However, existing tools for comparing distributions are slow and costly as datasets grow large. This project develops new geometry-aware methods for comparing distributional data that are also computationally efficient, enabling fast feature extraction and preprocessing at scale. This project addresses a fundamental step in distributional data analysis and contributes to scalable and trustworthy machine learning. More broadly, the project delivers distributional analysis tools that are applicable across scientific domains and supports more efficient technologies in artificial intelligence, an area of national strategic priority, that enhance innovation, strengthen economic competitiveness, and produce societal benefits. The project also supports education and workforce development through graduate student training, curriculum design, and public engagement. This project introduces and analyzes a family of distributional metrics induced by a joint extension of stochastic localization, a technique that has had a significant impact on recent advances in convex geometry and score-based generative models. The investigator studies these distributional metrics from three perspectives: theoretical, computational, and applied. Theoretically, the investigator characterizes the mathematical properties of these metrics, including how the topologies they induce compare with those induced by existing distributional metrics, their geodesic structures, and the summary statistics that describe the resulting geometries. Computationally, the investigator designs efficient algorithms for computing these metrics, establishes their convergence guarantees, and builds distributional embedding tools based on these algorithms for data processing. From an applied perspective, these theoretical and computational developments are integrated to address problems in machine learning, such as distribution learning and surrogate modeling for optimal transport. The work draws on and connects ideas from probability theory, statistics, convex geometry, scientific computing, and optimal transport. The expected outcome of this project is a set of mathematically grounded, computationally efficient tools for distributional data analysis and, more generally, scalable machine learning. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.