Thesis (M.A.) -- Memorial University of Newfoundland. / Typescript. Bibliography : leaves 62-64. Also available online.

Thesis (M.S.)--Oregon State University, 2010. / Printout. Includes bibliographical references (leaves 67-71). Also available on the World Wide Web.

A general approach to the treatment of electrical machine systems is developed. Tensor concepts are adopted; however, metrical ideas are avoided in favour of Hamilton's Principle. Using Lie derivatives and choosing a holonomic reference system, the equations resulting are general, and thus apply to any physical system of machines. These equations are Faraday's Law for the electrical portion and a gradient equation for the mechanical portion. Transformation characteristics, which are found to be of two independent types, called the v-type and the i-type,are investigated. This leads to tensor character and invariance properties associated with the transformations. The equations of small oscillation, which are based on the general equations of motion obtained in the thesis, are derived for any physical system. In the final chapter two examples of application are given; the power selsyn system, and the synchronous machine. / Applied Science, Faculty of / Electrical and Computer Engineering, Department of / Graduate

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.

<p>Unlike matrix rank, hypermatrix rank is not lower semi-continuous. As a result, optimal low rank approximations of hypermatrices may not exist. Characterizing hypermatrices without optimal low rank approximations is an important step in implementing algorithms with hypermatrices. The main result of this thesis is an original coordinate-free proof that real 2 by 2 by 2 tensors that are rank three do not have optimal rank two approximations with respect to the Frobenius norm. This result was previously only proved in coordinates. Our coordinate-free proof expands on prior results by developing a proof method that can be generalized more readily to higher dimensional tensor spaces. Our proof has the corollary that the nearest point of a rank three tensor to the second secant set of the Segre variety is a rank three tensor in the tangent space of the Segre variety. The relationship between the contraction maps of a tensor generalizes, in a coordinate-free way, the fundamental relationship between the rows and columns of a matrix to hypermatrices. Our proof method demonstrates geometrically the fundamental relationship between the contraction maps of a tensor. For example, we show that a regular real or complex tensor is tangent to the 2 by 2 by 2 Segre variety if and only if the image of any of its contraction maps is tangent to the 2 by 2 Segre variety. </p>

Neural data often span multiple indices, such as neuron, experimental condition, trial, and time, resulting in a tensor or multidimensional array. Standard approaches to neural data analysis often rely on matrix factorization techniques, such as principal component analysis or nonnegative matrix factorization. Any inherent tensor structure in the data is lost when flattened into a matrix. Here, we analyze datasets from primary motor cortex from the perspective of tensor analysis, and develop a theory for how tensor structure relates to certain computational properties of the underlying system. Applied to the motor cortex datasets, we reveal that neural activity is best described by condition-independent dynamics as opposed to condition-dependent relations to external movement variables. Motivated by this result, we pursue one further tensor-related analysis, and two further dynamical systems-related analyses. First, we show how tensor decompositions can be used to denoise neural signals. Second, we apply system identification to the cortex- to-muscle transformation to reveal the intermediate spinal dynamics. Third, we fit recurrent neural networks to muscle activations and show that the geometric properties observed in motor cortex are naturally recapitulated in the network model. Taken together, these results emphasize (on the data analysis side) the role of tensor structure in data and (on the theoretical side) the role of motor cortex as a dynamical system.

Tensors, a.k.a. multi-dimensional arrays, arise naturally when modeling higher-order objects and relations. Among ubiquitous applications including image processing, collaborative filtering, demand forecasting and higher-order statistics, there are two recurring themes in general: tensor recovery and tensor decomposition. The first one aims to recover the underlying tensor from incomplete information; the second one is to study a variety of tensor decompositions to represent the array more concisely and moreover to capture the salient characteristics of the underlying data. Both topics are respectively addressed in this thesis. Chapter 2 and Chapter 3 focus on low-rank tensor recovery (LRTR) from both theoretical and algorithmic perspectives. In Chapter 2, we first provide a negative result to the sum of nuclear norms (SNN) model---an existing convex model widely used for LRTR; then we propose a novel convex model and prove this new model is better than the SNN model in terms of the number of measurements required to recover the underlying low-rank tensor. In Chapter 3, we first build up the connection between robust low-rank tensor recovery and the compressive principle component pursuit (CPCP), a convex model for robust low-rank matrix recovery. Then we focus on developing convergent and scalable optimization methods to solve the CPCP problem. In specific, our convergent method, proposed by combining classical ideas from Frank-Wolfe and proximal methods, achieves scalability with linear per-iteration cost. Chapter 4 generalizes the successive rank-one approximation (SROA) scheme for matrix eigen-decomposition to a special class of tensors called symmetric and orthogonally decomposable (SOD) tensor. We prove that the SROA scheme can robustly recover the symmetric canonical decomposition of the underlying SOD tensor even in the presence of noise. Perturbation bounds, which can be regarded as a higher-order generalization of the Davis-Kahan theorem, are provided in terms of the noise magnitude.

Thesis (Ph. D.)--University of Washington, 2002. / Vita. Includes bibliographical references (p. 57-59).

Thesis (M. Phil.)--Hong Kong University of Science and Technology, 2003. / Includes bibliographical references (leaves 123-127). Also available in electronic version. Access restricted to campus users.

Thesis (Ph.D..(Mathematics and Applied Mathematics)) -- University of Pretoria, 2009. / Summary in English. Includes bibliographical references.