Exploiting Low-dimensional Structure and Optimal Transport for Tracking and Alignment

Committee:

Dr. Christopher Rozell, ECE, Chair , Advisor

Dr. Justin Romberg, ECE

Dr. Eva Dyer, BME

Dr. Mark Davenport, ECE

Dr. Craig Forest, ME

Abstract:

The objective of this thesis is to exploit low-dimensional structures (e.g., sparsity) and optimal transport theory to develop new tools for inference and distribution alignment problems. We investigate properties of structure at two scales: local structure of the single datum, and global structure across the dataset's entirety. To study local notions of structure, we consider the fundamental problem of support mismatch under the framework of signal inference: inference suffers when the signal support is poorly estimated. Popular metrics (e.g., Lp-norms) are particularly prone to mismatch due to its lack of machinery to describe geometric correlations between support locations. To fill this gap, we exploit optimal transport theory to propose a dynamical regularizer that "understands" geometry. In addition, we develop efficient methods to overcome the traditionally-prohibitive costs of using optimal transport in large-scale applications. To understand global notions of structure, we consider the challenging problem of distribution alignment, which spans fields such as machine learning, computer vision, and graph matching. To bypass the intractability of graph matching approaches, we approach this problem from a machine learning perspective and exploit statistical advantages of optimal transport to align distributions. We develop methods that incorporate manifold and cluster structures that are necessary to regularize against convergence to poor local-minima and demonstrate the superiority of our method on synthetic and real data. Finally, we present pioneering results in cluster-based alignability analysis, which gives us theoretical conditions on when datasets can be aligned, as well as error bounds when the alignment transformation is isometric.