Theses and Dissertations
This collection includes both ASU Theses and Dissertations, submitted by graduate students, and the Barrett, Honors College theses submitted by undergraduate students.
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- Creators: Turaga, Pavan
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
The field of Computer Vision has seen great accomplishments in the last decade due to the advancements in Deep Learning. With the advent of Convolutional Neural Networks, the task of image classification has achieved unimaginable success when perceived through the traditional Computer Vision lens. With that being said, the state-of-the-art results in the image classification task were produced under a closed set assumption i.e. the input samples and the target datasets have knowledge of class labels in the testing phase. When any real-world scenario is considered, the model encounters unknown instances in the data. The task of identifying these unknown instances is called Open-Set Classification. This dissertation talks about the detection of unknown classes and the classification of the known classes. The problem is approached by using a neural network architecture called Deep Hierarchical Reconstruction Nets (DHRNets). It is dealt with by leveraging the reconstruction part of the DHRNets to identify the known class labels from the data. Experiments were also conducted on Convolutional Neural Networks (CNN) on the basis of softmax probability, Autoencoders on the basis of reconstruction loss, and Mahalanobis distance on CNN's to approach this problem.
ContributorsAinala, Kalyan (Author) / Turaga, Pavan (Thesis advisor) / Moraffah, Bahman (Committee member) / Demakethepalli Venkateswara, Hemanth Kumar (Committee member) / Arizona State University (Publisher)
Created2021