Graduate

No description available.

<p> Personalized PageRank with high teleportation probability enables exploring the environment of a seed. With this insight, one can use an orthogonal factorization of a set of personalized PageRank vectors, like SVD, to derive a 2-dimensional representation of the network. This can be done for the whole network or a smaller piece. The power of this method lies in the fact that only a few columns, compared to the size of the networks, can be used to generate a local representation of the part of the network we are interested in. This technique has the potential to be seamlessly used for higher order structures, such as hypergraphs which have found a great deal of use for real-world data. This work investigates the characteristics of personalized PageRank and how it compares to the transition probabilities on the graph in terms of their ability to develop low dimensional representations. A key focus of the thesis are the similarities between the embeddings generated due to PageRank and those generated by spectral methods.</p>

With the increased presence of image-based data in modern applications, the need for robust methods of image denoising grows greater. The work presented herein considers two of the most ubiquitous approaches towards image denoising: variational and non-local methods. The effectiveness of these methods is assessed using quantitatively using peak signal-to-noise ratio and structural similarity index measure metrics. This study employs ��−means clustering, an unsupervised machine learning algorithm, to isolate the most dominant cluster centroids within the incoming data and propose the introduction of a new adaptive parameter into the non-local means framework. Motivated by the fact that a majority of discrepancies between clean and denoised images occur at feature edges, this study examines several convolution-based edge detection methods to isolate relevant feature. The resultant gradient and edge information is used to further parameterize the ��−means non-local method. An additional hybrid method involving the combined contributions of variational and ��−means non-local denoising is proposed, with the weighting determined by edge intensities. This method outperforms the other methods outlined in the study, both conventional and newly presented.

Numerical methods for solving initial value problems in ordinary differential equations are studied. A notation is introduced to represent cyclic methods in terms of two matrices, A_h, and B_h, and this is developed to cover the very extensive class of m-block methods. Some stability results are obtained and convergence is analysed by means of a new consistency concept, namely optimal consistency. It is shown that optimal consistency allows one to give two-sided bounds on the global error, and examples are given to illustrate this. The form of the inverse of A_h is studied closely to give a criterion for the order of convergence to exceed that of consistency by one. Further convergence results are obtained , the first of which gives the orders of convergence for cases in which A_h, and B_h, have a special form, and the second of which gives rise to the possibility of the order of convergence exceeding that of consistency by two or more at some stages. In addition an alternative proof is given of the superconvergence result for collocation methods. In conclusion the work covered is set in the context of that done in recent years by various authors.

published_or_final_version / abstract / Mathematics / Master / Master of Philosophy

published_or_final_version / Computer Science / Doctoral / Doctor of Philosophy

No description available.

No description available.

In this thesis, a new smoothing penalty algorithm is introduced to solve a mathematical program with equilibrium constraints (MPEC). By smoothing the exact penalty function, an MPEC is reformulated as a series of subprograms which belong to a class of MPECs with simple linear complementarity constraints. To deal with the subproblems, a hybrid algorithm is proposed, which combines the active set algorithm, the 6-active search algorithm and the PSQP algorithm. It is shown that the smoothing penalty algorithm converges globally to a M-stationary point of MPEC under weak conditions. Supervisor: Dr. Jane Ye (Department of Mathematics and Statistics) Co-Supervisor: Dr. Wu-Sheng Lu (Department of Electrical and Computer Engineering)