The study of metric spaces is closely related to the study of topology in that the study of metric spaces concerns itself, also, with sets of points and with a limit point concept based on a function which gives a "distance" between two points. In some topological spaces it is possible to define a distance function between points in such a way that a limit point of a set in the topological sense is also a limit point of the same set in a metric sense. In such a case the topological space is "metrizable". The real numbers with its usual topology is an example of a topological space which is metrizable, the distance function being the absolute value of the difference of two real numbers. Chapters II and III of this thesis attempt to classify, to a certain extent, what type of topological space is metrizable. Chapters IV and V deal with several properties of metric spaces and certain functions of metric spaces, respectively.

In a large class of space-times, the specification of the curvature tensor components Rabcd in some coordinate domain of the space-time uniquely determines the metric up to a constant conformal factor. The purpose of this thesis is to investigate the spaces where the metric is not so determined, and to look at the determination of the metric when the components of the derivatives of the Riemann tensor (one index up) are also specified, with special reference to the role of the infinitesimal holonomy group (ihg). In chapter one we set up the mathematical background, describing the Weyl and Ricci tensor classifications, and defining holonomy. In chapter two we look at spaces with Riemann tensors of low rank. This leads us on to decomposable spaces and the connection between decomposable spaces and relativity in three dimensions. We examine the connection between decomposability and the ihg, and relate this to the Weyl and Ricci tensor classifications. In chapter three we discuss the problem of determination of the metric by the Riemann tensor alone, and give a brief review of the history of the problem. In chapter four we go on to look at the determination of the metric by the curvature and its derivatives. It is shown that, with the exception of the generalised pp-waves, we only need look as far as the first derivatives of the Riemann tensor to obtain the best determination of the metric, unless the Riemann tensor is rank 1, when the second derivatives may also be required. The form of the metric ambiguity, the ihg and Petrov types are determined in each case. These results are then reviewed in the final chapter.

by Hung Ching Nam. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaf 100). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 2 --- Sobolev spaces of maps into metric space --- p.9 / Chapter 2.1 --- "Lp(Ω,X) spaces" --- p.9 / Chapter 2.2 --- Maps with finite energy --- p.11 / Chapter 2.3 --- Differentiation of maps along a direction --- p.28 / Chapter 2.4 --- Theory of differentiation of maps --- p.35 / Chapter 2.5 --- Trace of maps on Lipschitz domains --- p.48 / Chapter 3 --- Sobolev maps into NPC space --- p.58 / Chapter 3.1 --- NPC space --- p.58 / Chapter 3.2 --- NPC space with curvature bound --- p.69 / Chapter 3.3 --- Sobolev maps into NPC space --- p.71 / Chapter 3.4 --- Tensor inequality for Sobolev maps --- p.77 / Chapter 4 --- Harmonic maps into NPC space --- p.79 / Chapter 4.1 --- Existence and uniqueness of Dirichlet problem --- p.79 / Chapter 4.2 --- Interior Lipschitz continuity of harmonic maps --- p.81 / Chapter 5 --- Equivariant harmonic maps --- p.86 / Chapter 5.1 --- A functional analysis lemma --- p.86 / Chapter 5.2 --- Existence of equivariant harmonic maps --- p.87 / Chapter 5.3 --- Compactification of NPC space --- p.93 / Chapter 5.4 --- Isometric action on CAT(-l) space --- p.96

Similarity search is a task fundamental to many machine learning and data analytics applications, where distance metric learning plays an important role. However, since modern online applications continuously produce objects with new characteristics which tend to change over time, state-of-the-art similarity search using distance metric learning methods tends to fail when deployed in such applications without taking the change into consideration. In this work, we propose a Distance Metric Learning-based Continuous Similarity Search approach (CSS for short) to account for the dynamic nature of such data. CSS system adopts an online metric learning model to achieve distance metric evolving to adapt the dynamic nature of continuous data without large latency. To improve the accuracy of online metric learning model, a compact labeled dataset which is representative of the updated data is dynamically updated. Also, to accelerate similarity search, CSS includes an online maintained Locality Sensitive Hashing index to accelerate the similarity search. One, our labeled data update strategy progressively enriches the labeled data to assure continued representativeness, yet without excessively growing its size to ensure that the computation costs of metric learning remain bounded. Two, our continuous distance metric learning strategy ensures that each update only requires one linear time k-NN search in contrast to the cubic time complexity of relearning the distance metric from scratch. Three, our LSH update mechanism leverages our theoretical insight that the LSH built based on the original distance metric is equally effective in supporting similarity search using the new distance metric as long as the transform matrix learned for the new distance metric is reversible. This important observation empowers CSS to avoid the modification of LSH in most cases. Our experimental study using real-world public datasets and large synthetic datasets confirms the effectiveness of CSS in improving the accuracy of classification and information retrieval tasks. Also, CSS achieves 3 orders of magnitude speedup of our incremental distance metric learning strategy (and its three underlying components) over the state-of-art methods.

Li Yuanyuan. / "July 2000." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 126-130). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Habilitationsschrift--Bonn. / Bibliography: p. 81.

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Thesis (M.S.)--Ohio State University, 2008. / Title from first page of PDF file. Includes bibliographical references (p. 22).

Thesis (M.C.S.) Carleton University, 2005. / Includes bibliographical references (p. 36-41). Also available in electronic format on the Internet.