Manifold Learning Theory and Applications

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Features

  • Describes the theoretical background as well as implementation and applications
  • Contains chapters written by some of the world's leading researchers and engineering professionals
  • Covers state-of-the-art literature in manifold learning
  • Includes 16 pages with color illustrations

Summary

Trained to extract actionable information from large volumes of high-dimensional data, engineers and scientists often have trouble isolating meaningful low-dimensional structures hidden in their high-dimensional observations. Manifold learning, a groundbreaking technique designed to tackle these issues of dimensionality reduction, finds widespread application in machine learning, neural networks, pattern recognition, image processing, and computer vision.

Filling a void in the literature, Manifold Learning Theory and Applications incorporates state-of-the-art techniques in manifold learning with a solid theoretical and practical treatment of the subject. Comprehensive in its coverage, this pioneering work explores this novel modality from algorithm creation to successful implementation—offering examples of applications in medical, biometrics, multimedia, and computer vision. Emphasizing implementation, it highlights the various permutations of manifold learning in industry including manifold optimization, large scale manifold learning, semidefinite programming for embedding, manifold models for signal acquisition, compression and processing, and multi scale manifold.

Beginning with an introduction to manifold learning theories and applications, the book includes discussions on the relevance to nonlinear dimensionality reduction, clustering, graph-based subspace learning, spectral learning and embedding, extensions, and multi-manifold modeling. It synergizes cross-domain knowledge for interdisciplinary instructions, offers a rich set of specialized topics contributed by expert professionals and researchers from a variety of fields. Finally, the book discusses specific algorithms and methodologies using case studies to apply manifold learning for real-world problems.

Table of Contents

Spectral Embedding Methods for Manifold Learning
Introduction
Spaces and Manifolds
Data on Manifolds
Linear Manifold Learning
Nonlinear Manifold Learning
Summary
Acknowledgment

Robust Laplacian Eigenmaps Using Global Information
Introduction
Graph Laplacian
Global Information of Manifold
Laplacian Eigenmaps with Global Information
Experiments
Summary
Bibliographical and Historical Remarks

Density Preserving Maps
Introduction
The Existence of Density Preserving Maps
Density Estimation on Submanifolds
Preserving the Estimated Density: The Optimization
Summary
Bibliographical and Historical Remarks

Sample Complexity in Manifold Learning
Introduction
Sample Complexity of Classification on a Manifold
Learning Smooth Class Boundaries
Sample Complexity of Testing the Manifold Hypothesis
Connections and Related Work
Sample Complexity of Empirical Risk Minimization
Relating Bounded Curvature to Covering Number
Class of Manifolds with a Bounded Covering Number
Fat-Shattering Dimension and Random Projections
Minimax Lower Bounds on the Sample Complexity
Algorithmic Implications
Summary

Manifold Alignment
Introduction
Formalization and Analysis
Variants of Manifold Alignment
Application Examples
Summary
Bibliographical and Historical Remarks
Acknowledgments

Large-scale Manifold Learning
Introduction
Background
Comparison of Sampling Methods
Large-Scale Manifold Learning
Summary
Bibliography and Historical Remarks

Metric and Heat Kernel
Introduction

Theoretic Background
Discrete Heat Kernel
Heat Kernel Simplification
Numerical Experiments
Applications
Summary
Bibliographical and Historical Remarks

Discrete Ricci Flow for Surface and 3-Manifold
Introduction
Theoretic Background
Surface Ricci Flow
3-Manifold Ricci Flow
Applications
Summary
Bibliographical and Historical Remarks

2D and 3D Objects Morphing Using Manifold Techniques
Introduction
Interpolation on Euclidean spaces
Generalization of Interpolation Algorithms on a Manifold M
Interpolation on SO(m)
Application: The Motion of a Rigid Object in Space
Interpolation on Shape Manifold
Examples of Fitting Curves on Shape Manifolds
Summary

Learning Image Manifolds from Local Features
Introduction
Joint Feature-Spatial Embedding
Solving the Out-Of-Sample Problem
From Feature Embedding to Image Embedding
Applications
Summary
Bibliographical and Historical remarks

Human Motion Analysis Applications of Manifold Learning
Introduction
Learning A Simple Motion Manifold
Factorized Generative Models
Generalized Style Factorization
Solving for Multiple Factors
Examples
Summary
Bibliographical and Historical remarks

Editor Bio(s)

About the Editors:

Yunqian Ma received his PhD in electrical engineering from the University of Minnesota at twin cities in 2003. He then joined Honeywell International Inc., where he is currently senior principal research scientist in the advanced technology lab at Honeywell Aerospace. He holds 12 U.S. patents and 38 patent applications. He has authored 50 publications, including 3 books. His research interest includes inertial navigation, integrated navigation, surveillance, signal and image processing, pattern recognition and computer vision, machine learning and neural networks. His research has been supported by internal funds and external contracts, such as AFRL, DARPA, HSARPA, and FAA. Dr. Ma received the International Neural Network Society (INNS) Young Investigator Award for outstanding contributions in the application of neural networks in 2006. He is currently associate editor of IEEE Transactions on Neural Networks, on the editorial board of the pattern recognition letters journal, and has served on the program committee of several international conferences. He also served on the panel of the National Science Foundation in the division of information and intelligent system and is a senior member of IEEE. Dr. Ma is included in Marquis Who is Who Engineering and Science.

Yun Fu received his B.Eng. in information engineering and M.Eng. in pattern recognition and intelligence systems, both from Xian Jiaotong University, China. His M.S. in statistics, and Ph.D. in electrical and computer engineering, were both earned at the University of Illinois at Urbana-Champaign. He joined BBN Technologies, Cambridge, MA, as a Scientist in 2008 and was a part-time lecturer with the Department of Computer Science, Tufts University, Medford, MA, in 2009. Since 2010, he has been an assistant professor with the Department of Computer Science and Engineering, SUNY at Buffalo. His current research interests include applied machine learning, human-centered computing, pattern recognition, intelligent vision system, and social media analysis. Dr. Fu is the recipient of the 2002 Rockwell Automation Master of Science Award, Edison Cups of the 2002 GE Fund Edison Cup Technology Innovation Competition, the 2003 Hewlett-Packard Silver Medal and Science Scholarship, the 2007 Chinese Government Award for Outstanding Self-Financed Students Abroad, the 2007 DoCoMo USA Labs Innovative Paper Award (IEEE International Conference on Image Processing 2007 Best Paper Award), the 2007-2008 Beckman Graduate Fellowship, the 2008 M. E. Van Valkenburg Graduate Research Award, the ITESOFT Best Paper Award of 2010 IAPR International Conferences on the Frontiers of Handwriting Recognition (ICFHR), and the 2010 Google Faculty Research Award. He is a lifetime member of Institute of Mathematical Statistics (IMS), senior member of IEEE, member of ACM and SPIE.