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Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued

Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations.
ContributorsRobinson, Benjamin (Author) / Cochran, Douglas (Thesis advisor) / Moran, William (Thesis advisor) / Boggess, Albert (Committee member) / Milner, Fabio (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2014
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Spectral congestion is quickly becoming a problem for the telecommunications sector. In order to alleviate spectral congestion and achieve electromagnetic radio frequency (RF) convergence, communications and radar systems are increasingly encouraged to share bandwidth. In direct opposition to the traditional spectrum sharing approach between radar and communications systems of complete

Spectral congestion is quickly becoming a problem for the telecommunications sector. In order to alleviate spectral congestion and achieve electromagnetic radio frequency (RF) convergence, communications and radar systems are increasingly encouraged to share bandwidth. In direct opposition to the traditional spectrum sharing approach between radar and communications systems of complete isolation (temporal, spectral or spatial), both systems can be jointly co-designed from the ground up to maximize their joint performance for mutual benefit. In order to properly characterize and understand cooperative spectrum sharing between radar and communications systems, the fundamental limits on performance of a cooperative radar-communications system are investigated. To facilitate this investigation, performance metrics are chosen in this dissertation that allow radar and communications to be compared on the same scale. To that effect, information is chosen as the performance metric and an information theoretic radar performance metric compatible with the communications data rate, the radar estimation rate, is developed. The estimation rate measures the amount of information learned by illuminating a target. With the development of the estimation rate, standard multi-user communications performance bounds are extended with joint radar-communications users to produce bounds on the performance of a joint radar-communications system. System performance for variations of the standard spectrum sharing problem defined in this dissertation are investigated, and inner bounds on performance are extended to account for the effect of continuous radar waveform optimization, multiple radar targets, clutter, phase noise, and radar detection. A detailed interpretation of the estimation rate and a brief discussion on how to use these performance bounds to select an optimal operating point and achieve RF convergence are provided.
ContributorsChiriyath, Alex Rajan (Author) / Bliss, Daniel W (Thesis advisor) / Cochran, Douglas (Committee member) / Kosut, Oliver (Committee member) / Richmond, Christ D (Committee member) / Arizona State University (Publisher)
Created2018
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Description
Deconvolution of noisy data is an ill-posed problem, and requires some form of regularization to stabilize its solution. Tikhonov regularization is the most common method used, but it depends on the choice of a regularization parameter λ which must generally be estimated using one of several common methods. These methods

Deconvolution of noisy data is an ill-posed problem, and requires some form of regularization to stabilize its solution. Tikhonov regularization is the most common method used, but it depends on the choice of a regularization parameter λ which must generally be estimated using one of several common methods. These methods can be computationally intensive, so I consider their behavior when only a portion of the sampled data is used. I show that the results of these methods converge as the sampling resolution increases, and use this to suggest a method of downsampling to estimate λ. I then present numerical results showing that this method can be feasible, and propose future avenues of inquiry.
ContributorsHansen, Jakob Kristian (Author) / Renaut, Rosemary (Thesis director) / Cochran, Douglas (Committee member) / Barrett, The Honors College (Contributor) / School of Music (Contributor) / Economics Program in CLAS (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2015-05
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Description
The data explosion in the past decade is in part due to the widespread use of rich sensors that measure various physical phenomenon -- gyroscopes that measure orientation in phones and fitness devices, the Microsoft Kinect which measures depth information, etc. A typical application requires inferring the underlying physical phenomenon

The data explosion in the past decade is in part due to the widespread use of rich sensors that measure various physical phenomenon -- gyroscopes that measure orientation in phones and fitness devices, the Microsoft Kinect which measures depth information, etc. A typical application requires inferring the underlying physical phenomenon from data, which is done using machine learning. A fundamental assumption in training models is that the data is Euclidean, i.e. the metric is the standard Euclidean distance governed by the L-2 norm. However in many cases this assumption is violated, when the data lies on non Euclidean spaces such as Riemannian manifolds. While the underlying geometry accounts for the non-linearity, accurate analysis of human activity also requires temporal information to be taken into account. Human movement has a natural interpretation as a trajectory on the underlying feature manifold, as it evolves smoothly in time. A commonly occurring theme in many emerging problems is the need to \emph{represent, compare, and manipulate} such trajectories in a manner that respects the geometric constraints. This dissertation is a comprehensive treatise on modeling Riemannian trajectories to understand and exploit their statistical and dynamical properties. Such properties allow us to formulate novel representations for Riemannian trajectories. For example, the physical constraints on human movement are rarely considered, which results in an unnecessarily large space of features, making search, classification and other applications more complicated. Exploiting statistical properties can help us understand the \emph{true} space of such trajectories. In applications such as stroke rehabilitation where there is a need to differentiate between very similar kinds of movement, dynamical properties can be much more effective. In this regard, we propose a generalization to the Lyapunov exponent to Riemannian manifolds and show its effectiveness for human activity analysis. The theory developed in this thesis naturally leads to several benefits in areas such as data mining, compression, dimensionality reduction, classification, and regression.
ContributorsAnirudh, Rushil (Author) / Turaga, Pavan (Thesis advisor) / Cochran, Douglas (Committee member) / Runger, George C. (Committee member) / Taylor, Thomas (Committee member) / Arizona State University (Publisher)
Created2016
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Description
This thesis develops geometrically and statistically rigorous foundations for multivariate analysis and bayesian inference posed on grassmannian manifolds. Requisite to the development of key elements of statistical theory in a geometric realm are closed-form, analytic expressions for many differential geometric objects, e.g., tangent vectors, metrics, geodesics, volume forms. The first

This thesis develops geometrically and statistically rigorous foundations for multivariate analysis and bayesian inference posed on grassmannian manifolds. Requisite to the development of key elements of statistical theory in a geometric realm are closed-form, analytic expressions for many differential geometric objects, e.g., tangent vectors, metrics, geodesics, volume forms. The first part of this thesis is devoted to a mathematical exposition of these. In particular, it leverages the classical work of Alan James to derive the exterior calculus of differential forms on special grassmannians for invariant measures with respect to which integration is permissible. Motivated by various multi-­sensor remote sensing applications, the second part of this thesis describes the problem of recursively estimating the state of a dynamical system propagating on the Grassmann manifold. Fundamental to the bayesian treatment of this problem is the choice of a suitable probability distribution to a priori model the state. Using the Method of Maximum Entropy, a derivation of maximum-­entropy probability distributions on the state space that uses the developed geometric theory is characterized. Statistical analyses of these distributions, including parameter estimation, are also presented. These probability distributions and the statistical analysis thereof are original contributions. Using the bayesian framework, two recursive estimation algorithms, both of which rely on noisy measurements on (special cases of) the Grassmann manifold, are the devised and implemented numerically. The first is applied to an idealized scenario, the second to a more practically motivated scenario. The novelty of both of these algorithms lies in the use of thederived maximum­entropy probability measures as models for the priors. Numerical simulations demonstrate that, under mild assumptions, both estimation algorithms produce accurate and statistically meaningful outputs. This thesis aims to chart the interface between differential geometry and statistical signal processing. It is my deepest hope that the geometric-statistical approach underlying this work facilitates and encourages the development of new theories and new computational methods in geometry. Application of these, in turn, will bring new insights and bettersolutions to a number of extant and emerging problems in signal processing.
ContributorsCrider, Lauren N (Author) / Cochran, Douglas (Thesis advisor) / Kotschwar, Brett (Committee member) / Scharf, Louis (Committee member) / Taylor, Thomas (Committee member) / Turaga, Pavan (Committee member) / Arizona State University (Publisher)
Created2021
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Description
During the inversion of discrete linear systems, noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion. This is a process called regularization. The influence of the provided prior information is controlled by

During the inversion of discrete linear systems, noise in data can be amplified and result in meaningless solutions. To combat this effect, characteristics of solutions that are considered desirable are mathematically implemented during inversion. This is a process called regularization. The influence of the provided prior information is controlled by the introduction of non-negative regularization parameter(s). Many methods are available for both the selection of appropriate regularization parame- ters and the inversion of the discrete linear system. Generally, for a single problem there is just one regularization parameter. Here, a learning approach is considered to identify a single regularization parameter based on the use of multiple data sets de- scribed by a linear system with a common model matrix. The situation with multiple regularization parameters that weight different spectral components of the solution is considered as well. To obtain these multiple parameters, standard methods are modified for identifying the optimal regularization parameters. Modifications of the unbiased predictive risk estimation, generalized cross validation, and the discrepancy principle are derived for finding spectral windowing regularization parameters. These estimators are extended for finding the regularization parameters when multiple data sets with common system matrices are available. Statistical analysis of these estima- tors is conducted for real and complex transformations of data. It is demonstrated that spectral windowing regularization parameters can be learned from these new esti- mators applied for multiple data and with multiple windows. Numerical experiments evaluating these new methods demonstrate that these modified methods, which do not require the use of true data for learning regularization parameters, are effective and efficient, and perform comparably to a supervised learning method based on es- timating the parameters using true data. The theoretical developments are validated for one and two dimensional image deblurring. It is verified that the obtained estimates of spectral windowing regularization parameters can be used effectively on validation data sets that are separate from the training data, and do not require known data.
ContributorsByrne, Michael John (Author) / Renaut, Rosemary (Thesis advisor) / Cochran, Douglas (Committee member) / Espanol, Malena (Committee member) / Jackiewicz, Zdzislaw (Committee member) / Platte, Rodrigo (Committee member) / Arizona State University (Publisher)
Created2023