CAREER: Robust Manifold and Metric Learning: Techniques for Noise, Intersections, and Geometric Regularization

This project aims to advance machine learning techniques by developing new mathematical tools to better analyze and visualize complex, high-dimensional data. Many real-world data sets, such as genetic information or molecular images, contain hidden structures that can be uncovered using geometric methods. Leveraging these hidden structures can decrease the computational resources needed to analyze large data sets and can also provide scientific insight regarding various biological and physical processes. The goal of this project is to harness the full power of geometric methods for modern machine learning by the design of innovative solutions to the critical challenges facing the field. The project will focus on developing methods that can handle noisy data and tools for data visualization that preserve important geometric details. In addition to its scientific goals, the project will have a broad impact by offering new educational programs to support student mental health, increase diversity in data science, and encourage underrepresented groups to engage in this field. By addressing both technical and social challenges, the project aims to create more reliable, scalable tools for data analysis while promoting inclusion and innovation in science and education. In many applications, real data often concentrates around low-dimensional structures or manifolds, and manifold learning algorithms are a powerful tool for uncovering this low-dimensional structure. The project is on manifold learning algorithms, tackling the key challenges facing the field, including how noise affects algorithm performance, how to preserve both local and global geometric details, and how to effectively handle complex collections of manifolds. This research will investigate best practices for denoising, analyze how noise impacts spectral manifold learning algorithms, and clarify the regime in which algorithms can reliably recover manifold structure. The project will develop novel algorithms that utilize diffusion to faithfully encode geometric information at various scales and also hybrid methods for geometric regularization of manifold learning algorithms. Since real data often fails to concentrate around a single manifold, this work will contribute a robust and scalable solution for clustering collections of manifolds via a novel angle-based path metric on simplices, so that manifold learning algorithms can be applied under less restrictive assumptions. In addition, since data often contains numerous irrelevant features and features measured on very different scales, the project aims to develop a probabilistic framework for learning and adjusting features according to their actual relevance. Overall, the goal is to design a comprehensive data analysis pipeline that is noise-robust and capable of balancing local and global geometric information to represent data in just a few dimensions. 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.