Geometric and Topological Reconstruction

Abstract

The understanding of mathematical signals is responsible for the information age. Computation, communication, and storage by computers all use signals, either implicitly or explicitly, and use mathematics to manipulate those signals. Reconstruction of a particular signal can be desirable or even necessary depending on how the signal manifests and is measured. We explore how to use mathematical ideas to manipulate and represent signals. Given measurements or samples or data, we analyze how to produce, or \emph{reconstruct}, the desired signal and the fundamental limits in doing so. We focus on reconstruction through a geometric and topological lens so that we can leverage geometric and topological constraints to solve the problems. As inaccuracies and noise are present in every computation, we adopt a statistical outlook and prove results with high probability given noise. We start off with probability and statistics and then use that for active reconstruction where the probability signal needs to be estimated statistically from sampling various sources. We prove optimal ways to doing this even in the most challenging of situations. Then we discuss functional analysis and how to reconstruct sparse rank one decompositions of operators. We prove optimality of certain matrix classes, based on geometry, and compute the worst case via sampling distributions. With the mathematical tools of functional analysis, we introduce the optimal transportation problem. Then we can use the Wasserstein metric and its geometry to provably reconstruct sparse signals with added noise. We devise an algorithm to solve this optimization problem and confirm its ability on both simulated data and real data. Heavily under-sampled data can be ill-posed which is often the case with magnetic resonance imaging data. We leverage the geometry of the motion correction problem to devise an appropriate approximation with a bound. Then we implement and confirm in simulation and on real data. Topology constraints are often present in non-obvious ways but can often be detected with persistent homology. We introduce the barcode algorithm and devise a method to parallelize it to allow analyzing large datasets. We prove the parallelization speedup and use it for natural language processing. We use topology constraints to reconstruct word-sense signals. Persistent homology is dependent on the data manifold, if it exists. And it is dependent on the manifold's reach. We calculate manifold reach and prove the instability of the formulation. We introduce the combinatorial reach to generalize reach and we prove the combinatorial reach is stable. We confirm this in simulation. Unfortunately, reach and persistent homology are not an invariant of hypergraphs. We discuss hypergraphs and how they can partially reconstruct joint distributions. We define a hypergraph and prove its ability to distinguish certain joint distributions. We give an approximation and prove its convergence. Then we confirm our results in simulation and prove its usefulness on a real dataset.