The Geometry of Signal and Image Patch-Sets

In this thesis, we study the representation of local, or fine scale, snippets --- or patches --- that are extracted from a signal or image. We describe a method that characterizes the dimensionality that is observed in the set of patches when they are regarded as points in Euclidean space. Our approach is based on the assumption that the signal or image is composed of solutions to ordinary differential equations of a certain class. We also provide a theoretical interpretation --- via graph models --- that explains the success of analyzing signal and image patches using diffusion-based graph metrics. Our framework is built on the assumption that there exists a partition of the signal or image\u27s patches. Specifically, we assume there are two subsets of patches. One set comprises patches that are connected through some type of coherence in the domain of the signal, such as temporal coherence in time series, or spatial coherence between patches in the image plane. The other set comprises patches whose edge connections are not so largely influenced by the aforementioned coherence. Instead, these connections are more sporadic, with little relationship between the locations in the signal or image domain from which the patches were extracted. Using the commute time metric --- a diffusion-based graph metric --- we prove that the average proximity between patches in the first set grows faster than the average proximity between patches in the second set, as the number of patches approaches infinity. Consequently, a parametrization of the patches based on commute times will relatively cluster the second set of patches, which is the first step toward solving a larger problem, such as classification or clustering of the patches, detection of anomalies, or segmentation of an image. In addition to our theoretical results, this thesis also evaluates numerical procedures designed to efficiently compute the spectral decomposition of large matrices. These procedures include the Nystrom extension, and a multilevel eigensolver. Finally, we benchmark a classifier that is trained on the commute time embedding of a dataset of seismic events, against a standard algorithm used to detect arrival-times

Chern-Simons theory of magnetization plateaus on the kagome lattice

Frustrated spin systems on Kagome lattices have long been considered to be a promising candidate for realizing exotic spin liquid phases. Recently, there has been a lot of renewed interest in these systems with the discovery of experimental materials such as Volborthite and Herbertsmithite that have Kagome like structures. In this thesis I will focus on studying frustrated spin systems on the Kagome lattice using a spin-1/2 antiferromagnetic XXZ Heisenberg model in the presence of an external magnetic field as well as other perturbations. Such a system is expected to give rise to magnetization platueaus which can exhibit topological characteristics in certain regimes. We will first develop a flux-attachment transformation that maps the Heisenberg spins (hard-core bosons) onto a problem of fermions coupled to a Chern-Simons gauge field. This mapping relies on being able to define a consistent Chern-Simons term on the lattice. Using this newly developed mapping we analyse the phases/magnetization plateaus that arise at the mean-field level and also consider the effects of adding fluctuations to various mean-fi eld states. Along the way, we show how to discretize an abelian Chern-Simons gauge theory on generic 2D planar lattices that satisfy certain conditions. We find that as long as there exists a one-to-one correspondence between the vertices and plaquettes defined on the graph, one can write down a discretized lattice version of the abelian Chern-Simons gauge theory. Using the newly developed flux attachment transformation, we show the existence of chiral spin liquid states for various magnetization plateaus for certain range of parameters in the XXZ Heisenberg model in the presence of an external magnetic field. Speci cally, in the regime of XY anisotropy the ground states at the 1/3 and 2/3 plateau are equivalent to a bosonic fractional quantum Hall Laughlin state with filling fraction 1/2 and that the 5/9 plateau is equivalent to the first bosonic Jain daughter state at filling fraction 2/3. Next, we also consider the effects of several perturbations: a) a chirality term, b) a Dzyaloshinskii-Moriya term, and c) a ring-exchange type term on the bowties of the kagome lattice, and inquire if they can also support chiral spin liquids as ground states. We find that the chirality term leads to a chiral spin liquid even in the absence of an uniform magnetic field, with an effective spin Hall conductance of 1/2 in the regime of XY anisotropy. The Dzyaloshinkii-Moriya term also leads a similar chiral spin liquid but only when this term is not too strong. An external magnetic field when combined with some of the above perturbations also has the possibility of giving rise to additional plateaus which also behave like chiral spin liquids in the XY regime. Under the in influence of a ring-exchange term we find that provided its coupling constant is large enough, it may trigger a phase transition into a chiral spin liquid by the spontaneous breaking of time-reversal invariance. Finally, we also present some numerical results based on some exact diagonalization studies. Here, we specifically focus on the 2/3-magnetization plateau which we previously argued should be a chiral spin liquid with a spin hall conductance of 1/2 . Such a topological state has a non-trivial ground state degeneracy and it excitations are described by semionic quasiparticles. In the numerical analysis, we analyse the ground state degeneracy structure on various Kagome clusters of different sizes. We compute modular matrices from the resultant minimally entangled states as well as the Chern numbers of various eigenstates all of which provide strong evidence that the 2/3-magnetization plateau very closely resembles a chiral spin liquid state with the expected characteristics