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Recursive Bayesian Estimation on Projective Spaces: Theoretical Foundations and Practical Algorithms
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 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
Description
Little is known about how cognitive and brain aging patterns differ in older adults with autism spectrum disorder (ASD). However, recent evidence suggests that individuals with ASD may be at greater risk of pathological aging conditions than their neurotypical (NT) counterparts. A growing body of research indicates that older adults with ASD may experience accelerated cognitive decline and neurodegeneration as they age, although studies are limited by their cross-sectional design in a population with strong age-cohort effects. Studying aging in ASD and identifying biomarkers to predict atypical aging is important because the population of older individuals with ASD is growing. Understanding the unique challenges faced as autistic adults age is necessary to develop treatments to improve quality of life and preserve independence. In this study, a longitudinal design was used to characterize cognitive and brain aging trajectories in ASD as a function of autistic trait severity. Principal components analysis (PCA) was used to derive a cognitive metric that best explains performance variability on tasks measuring memory ability and executive function. The slope of the integrated persistent feature (SIP) was used to quantify functional connectivity; the SIP is a novel, threshold-free graph theory metric which summarizes the speed of information diffusion in the brain. Longitudinal mixed models were using to predict cognitive and brain aging trajectories (measured via the SIP) as a function of autistic trait severity, sex, and their interaction. The sensitivity of the SIP was also compared with traditional graph theory metrics. It was hypothesized that older adults with ASD would experience accelerated cognitive and brain aging and furthermore, age-related changes in brain network topology would predict age-related changes in cognitive performance.
For both cognitive and brain aging, autistic traits and sex interacted to predict trajectories, such that older men with high autistic traits were most at risk for poorer outcomes. In men with autism, variability in SIP scores across time points trended toward predicting cognitive aging trajectories. Findings also suggested that autistic traits are more sensitive to differences in brain aging than diagnostic group and that the SIP is more sensitive to brain aging trajectories than other graph theory metrics. However, further research is required to determine how physiological biomarkers such as the SIP are associated with cognitive outcomes.
ContributorsSullivan, Georgia (Author) / Braden, Blair (Thesis advisor) / Kodibagkar, Vikram (Thesis advisor) / Schaefer, Sydney (Committee member) / Wang, Yalin (Committee member) / Arizona State University (Publisher)
Created2022
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 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
Description
Compressive sensing theory allows to sense and reconstruct signals/images with lower sampling rate than Nyquist rate. Applications in resource constrained environment stand to benefit from this theory, opening up many possibilities for new applications at the same time. The traditional inference pipeline for computer vision sequence reconstructing the image from compressive measurements. However,the reconstruction process is a computationally expensive step that also provides poor results at high compression rate. There have been several successful attempts to perform inference tasks directly on compressive measurements such as activity recognition. In this thesis, I am interested to tackle a more challenging vision problem - Visual question answering (VQA) without reconstructing the compressive images. I investigate the feasibility of this problem with a series of experiments, and I evaluate proposed methods on a VQA dataset and discuss promising results and direction for future work.
ContributorsHuang, Li-Chin (Author) / Turaga, Pavan (Thesis advisor) / Yang, Yezhou (Committee member) / Li, Baoxin (Committee member) / Arizona State University (Publisher)
Created2017