Exploiting low-dimensional structure and optimal transport for tracking and alignment

The objective of this thesis is to exploit low-dimensional structures (e.g., sparsity, low-rankness) 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 along the temporal continuum, 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 regularizers that explicitly incorporate geometry. To realize such regularizers at scale, we develop efficient methods to overcome the traditionally-prohibitive computational costs of computing optimal transport. To understand global notions of structure, we consider the challenging problem of distribution alignment, which spans fields of 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 when datasets can be aligned, as well as error bounds when the alignment transformation is constrained to be isometric.

Date

2019-08-26

Resource Type

Text

Resource Subtype

Dissertation

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