Optimal Transport Methods for Nonlinear Dimensionality Reduction and Applications
Abstract
One of the fundamental observations of modern data driven modeling is that high-dimensional data often exhibits low-dimensional structure. Detecting and utilizing structures such as sparsity, union of subspaces, or low-dimensional manifolds has been the driving force of innovation and success for many modern algorithms pertaining to image and video processing, clustering, and pattern recognition, and has led to better understanding of the success of neural network classifiers and other machine learning models. This proposal suggests a new paradigm which blends techniques from optimal transport theory and nonlinear dimensionality reduction to tackle challenging problems in the processing of imaging data. This project will result in new rigorous mathematical theory for nonlinear dimensionality reduction, including recovery guarantees of functional manifolds and discretizations thereof, and will also lead to a suite of new algorithms for dimensionality reduction which are applicable to imaging problems and more generally to any problem in which the data can be considered to be a positive integrable function, or the discretization of such. Potential applications of these methods are in image and video processing, and classification tasks on data which exhibits multiple manifold structures. The main goals of this project will be to develop and study a new paradigm for nonlinear dimensionality reduction and manifold learning by combining a functional manifold hypothesis (considering data as lying on a submanifold of a function space) and optimal transport theory (considering the functional manifold as a subset of the Wasserstein space of probability measures with finite second moment). Success of this project will lead to a suite of new nonlinear dimensionality reduction algorithms which are useful in various imaging applications, the only requirement of which is that each data point can be treated as a probability distribution over a given domain. Focus will be on a pipeline involving theoretical treatment of proposed algorithms, subsequent experimentation of these algorithms, and synthesizing these results into formulating new algorithms and methods which will then repeat the cycle.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jul 28, 2023
- Source ID
- W911NF2310213
Entities
People
- Keaton Hamm
Organizations
- Army Contracting Command
- United States Army
- University of Texas at Arlington