Thesis--Ph. D., University of Hong Kong, 1986.

published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

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This dissertation is motivated by the increasing applications of high-dimensional large-scale data sets in various fields and lack of theoretical understanding of the existing algorithms as well as lack of efficient algorithms in many cases. Hence, identifying the geometrical properties of data sets is essential for many data processing tasks, such as data retrieval and denoising. In Part I, we derive the fundamental limits on the sampling rate required to study three important problems (i) low-rank data completion, (ii) rank estimation, and (iii) data clustering. In Chapter 2 we characterize the geometrical conditions on the sampling pattern, i.e., locations of the sampled entries, for finite and unique completability of a low-rank tensor, assuming that its rank vector is given or estimated. To this end, we propose a manifold analysis and study the independence of a set of polynomials defined based on the sampling pattern. Then, using the polynomial analysis, we derive a lower bound on the sampling rate such that it guarantees that the proposed conditions on the sampling patterns for finite and unique completability hold true with high probability. Then, in Chapter 3, we study the problem of rank estimation, where a data structure is partially sampled and we propose a geometrical analysis on the sampling pattern to estimate the true value of rank for various data structures by providing extremely tight lower and upper bounds on the rank value. And in Chapters 4 and 5, we make use of the developed tools to obtain a lower bound on the sampling rate to be able to correctly cluster a union of sampled matrices or tensors by identifying their corresponding unknown subspaces. In Part II, first in Chapter 6, motivated by the algebraic tools developed in Part I, we develop a data completion algorithm based on solving a set of polynomial equations using Newton's method, that is effective especially when the sampling rate is low. Then, in Chapter 7, we consider a data structure consisting of a union of nested low-rank matrix or tensor subspaces, and develop a structured alternating minimization-based approach for completing such data, that is capable of taking advantage of multiple rank constraints simultaneously to achieve faster convergence and higher recovery accuracy.

The objective of this dissertation work is the development of a solid theoretical and algorithmic framework for three of the most important aspects of autonomous/artificialintelligence (AI) systems, namely data quality assurance, localization, and communications. In the era of AI and machine learning (ML), data reign supreme. During learning tasks, we need to ensure that the training data set is correct and complete. During operation, faulty data need to be discovered and dealt with to protect from -potentially catastrophic- system failures. With our research in data quality assurance, we develop new mathematical theory and algorithms for outlier-resistant decomposition of high-dimensional matrices (tensors) based on L1-norm principal-component analysis (PCA). L1-norm PCA has been proven to be resistant to irregular data-points and will drive critical real-world AI learning and autonomous systems operations in the future. At the same time, one of the most important tasks of autonomous systems is self-localization. In GPS-deprived environments, localization becomes a fundamental technical problem. State-of-the-art solutions frequently utilize power-hungry or expensive architectures, making them difficult to deploy. In this dissertation work, we develop and implement a robust, variable-precision localization technique for autonomous systems based on the direction-of-arrival (DoA) estimation theory, which is cost and power-efficient. Finally, communication between autonomous systems is paramount for mission success in many applications. In the era of 5G and beyond, smart spectrum utilization is key.. In this work, we develop physical (PHY) and medium-access-control (MAC) layer techniques that autonomously optimize spectrum usage and minimizes intra and internetwork interference. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2020. / FAU Electronic Theses and Dissertations Collection

Thesis (Ph.D.)--Auburn University, 2007. / Abstract. Includes bibliographic references (ℓ. 68)

There are two primary sources of motivation for the contents of this thesis. The first is an effort to generalize classical dilation theory, a brief history of which is given in Section 2.1. The second source of motivation is the study of the representation theory of tensor algebras associated to C*-correspondences; these concepts are discussed in Sections 2.2 and 2.4. Although seemingly unrelated, there is a close connection between these two motivating theories. The link between classical dilation theory and the representation theory of tensor algebras over C*-correspondences was established by Muhly and Solel in their 1998 paper Tensor Algebras over C*-Correspondences: Representations, Dilations, and C*-Envelopes. In that paper, the authors not only introduced the concept of (operator-theoretic) tensor algebras – non-selfadjoint operator algebras that generalize algebraic tensor algebras – but they also developed the representation theory of these algebras. In order to do so, they introduced and made extensive use of a generalized dilation theory for contractions on Hilbert space. In analogy with classical dilation theory, they developed notions of “isometric dilation” and “coisometric extension” for completely contractive representations of the tensor algebra. The process of forming isometric dilations proceeded smoothly, but constructing coisometric extensions proved more problematic. In contrast to the classical case, Muhly and Solel showed that there is a high degree of nonuniqueness involved when building coisometric extensions. This lack of uniqueness proved to be an impediment to developing a full generalization of the dilation and model theories of Sz.-Nagy and Foias. In this thesis, we introduce a way to manage the ambiguities that arise when forming coisometric extensions. More specifically, we show that the notion of a transfer operator from classical dynamics can be adapted to this setting, and we prove that when a transfer operator is fixed in advance, every completely contractive representation of the tensor algebra admits a unique coisometric extension that respects the transfer operator in a fashion that we describe in Chapter 5. We also prove a commutant lifting theorem in the context of coisometric extensions.

Thesis (Ph. D.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 140-144).

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