Andrew Ranicki
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic ...
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This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.Less
This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.
Andrew Ranicki
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.003.0001
- Subject:
- Mathematics, Geometry / Topology
This chapter provides an introduction to surgery theory, including the definition of manifold and Poincaré complex. It also discusses h- and s-cobordisms.
This chapter provides an introduction to surgery theory, including the definition of manifold and Poincaré complex. It also discusses h- and s-cobordisms.
Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien (eds)
- Published in print:
- 2006
- Published Online:
- August 2013
- ISBN:
- 9780262033589
- eISBN:
- 9780262255899
- Item type:
- book
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262033589.001.0001
- Subject:
- Computer Science, Machine Learning
In the field of machine learning, semi-supervised learning (SSL) occupies the middle ground, between supervised learning (in which all training examples are labeled) and unsupervised learning (in ...
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In the field of machine learning, semi-supervised learning (SSL) occupies the middle ground, between supervised learning (in which all training examples are labeled) and unsupervised learning (in which no label data are given). Interest in SSL has increased in recent years, particularly because of application domains in which unlabeled data are plentiful, such as images, text, and bioinformatics. This overview of SSL presents state-of-the-art algorithms, a taxonomy of the field, selected applications, benchmark experiments, and perspectives on ongoing and future research. It first presents the key assumptions and ideas underlying the field: smoothness, cluster or low-density separation, manifold structure, and transduction. The core of the book is the presentation of SSL methods, organized according to algorithmic strategies. After an examination of generative models, the book describes algorithms that implement the low-density separation assumption, graph-based methods, and algorithms which perform two-step learning. It then discusses SSL applications and offers guidelines for SSL practitioners by analyzing the results of benchmark experiments. Finally, the book looks at interesting directions for SSL research. It closes with a discussion of the relationship between semi-supervised learning and transduction.Less
In the field of machine learning, semi-supervised learning (SSL) occupies the middle ground, between supervised learning (in which all training examples are labeled) and unsupervised learning (in which no label data are given). Interest in SSL has increased in recent years, particularly because of application domains in which unlabeled data are plentiful, such as images, text, and bioinformatics. This overview of SSL presents state-of-the-art algorithms, a taxonomy of the field, selected applications, benchmark experiments, and perspectives on ongoing and future research. It first presents the key assumptions and ideas underlying the field: smoothness, cluster or low-density separation, manifold structure, and transduction. The core of the book is the presentation of SSL methods, organized according to algorithmic strategies. After an examination of generative models, the book describes algorithms that implement the low-density separation assumption, graph-based methods, and algorithms which perform two-step learning. It then discusses SSL applications and offers guidelines for SSL practitioners by analyzing the results of benchmark experiments. Finally, the book looks at interesting directions for SSL research. It closes with a discussion of the relationship between semi-supervised learning and transduction.
Orlitsky Alon
- Published in print:
- 2006
- Published Online:
- August 2013
- ISBN:
- 9780262033589
- eISBN:
- 9780262255899
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262033589.003.0017
- Subject:
- Computer Science, Machine Learning
This chapter discusses density-based metrics induced by Riemannian manifold structures. It presents asymptotically consistent methods to estimate and compute these metrics and present upper and lower ...
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This chapter discusses density-based metrics induced by Riemannian manifold structures. It presents asymptotically consistent methods to estimate and compute these metrics and present upper and lower bounds on their estimation and computation errors. Finally, it is discussed how these metrics can be used for semi-supervised learning and present experimental results. Learning algorithms use a notion of similarity between data points to make inferences. Semi-supervised algorithms assume that two points are similar to each other if they are connected by a high-density region of the unlabeled data. Apart from semi-supervised learning, such density-based distance metrics also have applications in clustering and nonlinear interpolation.Less
This chapter discusses density-based metrics induced by Riemannian manifold structures. It presents asymptotically consistent methods to estimate and compute these metrics and present upper and lower bounds on their estimation and computation errors. Finally, it is discussed how these metrics can be used for semi-supervised learning and present experimental results. Learning algorithms use a notion of similarity between data points to make inferences. Semi-supervised algorithms assume that two points are similar to each other if they are connected by a high-density region of the unlabeled data. Apart from semi-supervised learning, such density-based distance metrics also have applications in clustering and nonlinear interpolation.
