  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.
