Thesis (Ph. D.)--University of Wyoming, 2005. / Title from PDF title page (viewed on Feb. 22, 2008). Includes bibliographical references (p. 136-139).

Thesis--Rijksunivers teit te Utrecht. / Includes bibliographical references (p. 103-104).

The research contained in this thesis can be divided into two related, but distinct parts. The rst chapter deals with block Toeplitz operators de ned by rational matrix function symbols on discrete sequence spaces. Here we study sequences of operators that converge to the inverses of these Toeplitz operators via an invertibility result involving a special representation of the symbol of these block Toeplitz operators. The second part focuses on a special class of matrices generated by banded Toeplitz matrices, i.e., Toeplitz matrices with a nite amount of non-zero diagonals. The spectral theory of banded Toeplitz matrices is well developed, and applied to solve questions regarding the behaviour of the singular values of Toeplitz-generated matrices. In particular, we use the behaviour of the singular values to deduce bounds for the growth of the norm of the inverse of Toeplitz-generated matrices. In chapter 2, we use a special state-space representation of a rational matrix function on the unit circle to de ne a block Toeplitz operator on a discrete sequence space. A discrete Riccati equation can be associated with this representation which can be used to prove an invertibility theorem for these Toeplitz operators. Explicit formulas for the inverse of the Toeplitz operators are also derived that we use to de ne a sequence of operators that converge in norm to the inverse of the Toeplitz operator. The rate of this convergence, as well as that of a related Riccati di erence equation is also studied. We conclude with an algorithm for the inversion of the nite sections of block Toeplitz operators. Chapter 3 contains the main research contribution of this thesis. Here we derive sharp growth rates for the norms of the inverses of Toeplitz-generated matrices. These results are achieved by employing powerful theory related to the Avram-Parter theorem that describes the distribution of the singular values of banded Toeplitz matrices. The investigation is then extended to include the behaviour of the extreme and general singular values of Toeplitz-generated matrices. We conclude with Chapter 4, which sets out to answer a very speci c question regarding the singular vectors of a particular subclass of Toeplitz-generated matrices. The entries of each singular vector seems to be a permutation (up to sign) of the same set of real numbers. To arrive at an explanation for this phenomenon, explicit formulas are derived for the singular values of the banded Toeplitz matrices that serve as generators for the matrices in question. Some abstract algebra is also employed together with some results from the previous chapter to describe the permutation phenomenon. Explicit formulas are also shown to exist for the inverses of these particular Toeplitz-generated matrices as well as algorithms to calculate the norms and norms of the inverses. Finally, some additional results are compiled in an appendix. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2015

The research contained in this thesis can be divided into two related, but distinct parts. The rst chapter deals with block Toeplitz operators de ned by rational matrix function symbols on discrete sequence spaces. Here we study sequences of operators that converge to the inverses of these Toeplitz operators via an invertibility result involving a special representation of the symbol of these block Toeplitz operators. The second part focuses on a special class of matrices generated by banded Toeplitz matrices, i.e., Toeplitz matrices with a nite amount of non-zero diagonals. The spectral theory of banded Toeplitz matrices is well developed, and applied to solve questions regarding the behaviour of the singular values of Toeplitz-generated matrices. In particular, we use the behaviour of the singular values to deduce bounds for the growth of the norm of the inverse of Toeplitz-generated matrices. In chapter 2, we use a special state-space representation of a rational matrix function on the unit circle to de ne a block Toeplitz operator on a discrete sequence space. A discrete Riccati equation can be associated with this representation which can be used to prove an invertibility theorem for these Toeplitz operators. Explicit formulas for the inverse of the Toeplitz operators are also derived that we use to de ne a sequence of operators that converge in norm to the inverse of the Toeplitz operator. The rate of this convergence, as well as that of a related Riccati di erence equation is also studied. We conclude with an algorithm for the inversion of the nite sections of block Toeplitz operators. Chapter 3 contains the main research contribution of this thesis. Here we derive sharp growth rates for the norms of the inverses of Toeplitz-generated matrices. These results are achieved by employing powerful theory related to the Avram-Parter theorem that describes the distribution of the singular values of banded Toeplitz matrices. The investigation is then extended to include the behaviour of the extreme and general singular values of Toeplitz-generated matrices. We conclude with Chapter 4, which sets out to answer a very speci c question regarding the singular vectors of a particular subclass of Toeplitz-generated matrices. The entries of each singular vector seems to be a permutation (up to sign) of the same set of real numbers. To arrive at an explanation for this phenomenon, explicit formulas are derived for the singular values of the banded Toeplitz matrices that serve as generators for the matrices in question. Some abstract algebra is also employed together with some results from the previous chapter to describe the permutation phenomenon. Explicit formulas are also shown to exist for the inverses of these particular Toeplitz-generated matrices as well as algorithms to calculate the norms and norms of the inverses. Finally, some additional results are compiled in an appendix. / PhD (Mathematics), North-West University, Potchefstroom Campus, 2015

Data comes in many di erent shapes and sizes. In real life applications it is common that data we are studying has features that are of varied data types. This may include, numerical, categorical, and text. In order to be able to model this data with machine learning algorithms, it is required that the data is typically in numeric form. Therefore, for data that is not originally numerical, it must be transformed to be able to be used as input into these algorithms. Along with this transformation it is common that data we study has many features relative to the number of samples in the data. It is often desirable to reduce the number of features that are being trained in a model to eliminate noise and reduce time in training. This problem of high dimensionality can be approached through feature selection, feature extraction, or feature embedding. Feature selection seeks to identify the most essential variables in a dataset that will lead to a parsimonious model and high performing results, while feature extraction and embedding are techniques that utilize a mathematical transformation of the data into a represented space. As a byproduct of using a new representation, we are able to reduce the dimension greatly without sacri cing performance. Oftentimes, by using embedded features we observe a gain in performance. Though extraction and embedding methods may be powerful for isolated machine learning problems, they do not always generalize well. Therefore, we are motivated to illustrate a methodology that can be applied to any data type with little pre-processing. The methods we develop can be applied in unsupervised, supervised, incremental, and deep learning contexts. Using 28 benchmark datasets as examples which include di erent data types, we construct a framework that can be applied for general machine learning tasks. The techniques we develop contribute to the eld of dimension reduction and feature embedding. Using this framework, we make additional contributions to eigendecomposition by creating an objective matrix that includes three main vital components. The rst being a class partitioned row and feature product representation of one-hot encoded data. Secondarily, the derivation of a weighted adjacency matrix based on class label relationships. Finally, by the inner product of these aforementioned values, we are able to condition the one-hot encoded data generated from the original data prior to eigenvector decomposition. The use of class partitioning and adjacency enable subsequent projections of the data to be trained more e ectively when compared side-to-side to baseline algorithm performance. Along with this improved performance, we can adjust the dimension of the subsequent data arbitrarily. In addition, we also show how these dense vectors may be used in applications to order the features of generic data for deep learning. In this dissertation, we examine a general approach to dimension reduction and feature embedding that utilizes a class partitioned row and feature representation, a weighted approach to instance similarity, and an adjacency representation. This general approach has application to unsupervised, supervised, online, and deep learning. In our experiments of 28 benchmark datasets, we show signi cant performance gains in clustering, classi cation, and training time. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

Thesis (Ph.D)--Computing, Georgia Institute of Technology, 2009. / Committee Chair: Santosh Vempala; Committee Member: Adam Kalai; Committee Member: Haesun Park; Committee Member: Ravi Kannan; Committee Member: Vladimir Koltchinskii. Part of the SMARTech Electronic Thesis and Dissertation Collection.

This thesis presents an improved method to calculate renormalization constants in a regularization invariant momentum scheme using twisted boundary conditions. This enables us to simulate with momenta of arbitrary magnitude and a fixed direction. With this new technique, together with non-exceptional kinematics and volume sources, we are able to take a statistically and theoretically precise continuum limit. Thereafter, all the running of the operators with momentum scale is due to their anomalous dimension. We use this to develop a practical scheme for step scaling with off shell vertex functions. We develop the method on 16³ × 32 lattices to show the practicality of using small volume simulations to step scale to high momenta. We also use larger 24³×64 and 32³×64 lattices to compute renormalization constants very accurately. Combining these with previous analyses we are able to extract a precise value for the light and strange quark masses and the neutral kaon mixing parameter BK. We also analyse eigenvectors of the domain wall Dirac matrix. We develop a practical and cost effective way to compute eigenvectors using the implicitly restarted Lanczos method with Chebyshev acceleration. We show that calculating eigenvectors to accelerate propagator inversions is cost effective when as few as one or two propagators are required. We investigate the technique of low mode averaging (LMA) with eigenvectors of the domain wall matrix for the first time. We find that for low energy correlators, pions for example, LMA is very effective at reducing the statistical noise. We also calculated the η and η′ meson masses, which required evaluating disconnected correlation functions and combining stochastic sources with LMA.

Thesis (Ph. D.)--Ohio State University, 2007. / Full text release at OhioLINK's ETD Center delayed at author's request

Thesis (Ph. D.)--Oregon State University, 2009. / Printout. Includes bibliographical references (leaves 298-303). Also available on the World Wide Web.

Thesis (Ph. D.)--University of Washington, 1994. / Vita. Includes bibliographical references (leaves [139]-141).