Computational Analysis of Deformable Manifolds: From Geometric Modeling to Deep Learning

Leo Tolstoy opened his monumental novel Anna Karenina with the now famous words: Happy families are all alike; every unhappy family is unhappy in its own way. A similar notion also applies to mathematical spaces: Every flat space is alike; every unflat space is unflat in its own way. However, rather than being a source of unhappiness, we will show that the diversity of non-flat spaces provides a rich area of study. The genesis of the so-called ’big data era’ and the proliferation of social and scientific databases of increasing size has led to a need for algorithms that can efficiently process, analyze and, even generate high dimensional data. However, the curse of dimensionality leads to the fact that many classical approaches do not scale well with respect to the size of these problems. One technique to avoid some of these ill-effects is to exploit the geometric structure of coherent data. In this thesis, we will explore geometric methods for shape processing and data analysis. More specifically, we will study techniques for representing manifolds and signals supported on them through a variety of mathematical tools including, but not limited to, computational differential geometry, variational PDE modeling and deep learning. First, we will explore non-isometric shape matching through variational modeling. Next, we will use ideas from parallel transport on manifolds to generalize convolution and convolutional neural networks to deformable manifolds. Finally, we conclude by proposing a novel auto-regressive model for capturing the intrinsic geometry and topology of data. Throughout this work, we will use the idea of computing correspondences as a though-line to both motivate our work and analyze our results. One of the advantages of working in this manner is that questions which arise from very specific problems will have far reaching consequences. There are many deep connections between concise models, harmonic analysis, geometry and learning that have only started to emerge in the past few years, and the consequences will continue to shape these fields for many years to come. Our goal in this work is to explore these connections and develop some useful tools for shape analysis, signal processing and representation learning