Algorithmic advances in Riemannian geometry and applications : for machine learning, computer vision, statistics, and optimization

216 Pages·2016·5.6 MB·English

by  Minh

Advances in Computer Vision and Pattern Recognition
Hà Quang Minh
Vittorio Murino Editors
Algorithmic
Advances in
Riemannian
Geometry and
Applications
For Machine Learning, Computer Vision,
Statistics, and Optimization
Advances in Computer Vision and Pattern
Recognition
Founding editor
Sameer Singh, Rail Vision, Castle Donington, UK
Series editor
Sing Bing Kang, Microsoft Research, Redmond, WA, USA
Advisory Board
Horst Bischof, Graz University of Technology, Austria
Richard Bowden, University of Surrey, Guildford, UK
Sven Dickinson, University of Toronto, ON, Canada
Jiaya Jia, The Chinese University of Hong Kong, Hong Kong
Kyoung Mu Lee, Seoul National University, South Korea
Yoichi Sato, The University of Tokyo, Japan
Bernt Schiele, Max Planck Institute for Computer Science, Saarbrücken, Germany
Stan Sclaroff, Boston University, MA, USA
More information about this series at http://www.springer.com/series/4205
à
H Quang Minh Vittorio Murino
(cid:129)
Editors
Algorithmic Advances
in Riemannian Geometry
and Applications
For Machine Learning, Computer Vision,
Statistics, and Optimization
123
Editors
HàQuangMinh Vittorio Murino
Pattern Analysis andComputer Vision Pattern Analysis andComputer Vision
Istituto Italiano diTecnologia Istituto Italiano diTecnologia
Genoa Genoa
Italy Italy
ISSN 2191-6586 ISSN 2191-6594 (electronic)
Advances in Computer Vision andPattern Recognition
ISBN978-3-319-45025-4 ISBN978-3-319-45026-1 (eBook)
DOI 10.1007/978-3-319-45026-1
LibraryofCongressControlNumber:2016948260
©SpringerInternationalPublishingSwitzerland2016
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Preface
Overview and Goals
The theme of this volume is the application of the rich and powerful theories and
techniquesofRiemanniangeometrytotheproblemsinmachinelearning,statistics,
optimization, computer vision, and related fields.
Traditional machine learning and data analysis methods often assume that the
inputdatacanberepresentedbyvectorsinEuclideanspace.Whilethisassumption
has worked well for many applications, researchers have increasingly realized that
if the data is intrinsically non-Euclidean, ignoring this geometrical structure can
lead to suboptimal results.
In the existing literature, there are two common approaches for exploiting data
geometry when the data is assumed to lie on a Riemannian manifold.
In the first direction, often referred to as manifold learning, the data is assumed
to lie on an unknown Riemannian manifold and the structure of this manifold is
exploited through the training data, either labeled or unlabeled. Examples of
manifold learning techniques include manifold regularization via the graph
Laplacian, locally linear embedding, and isometric mapping.
Intheseconddirection,whichisgainingincreasingimportanceandsuccess,the
Riemannianmanifoldrepresentingtheinputdataisassumedtobeknownexplicitly.
Somemanifoldsthathavebeenwidelyusedfordatarepresentationarethemanifold
of symmetric, positive definite matrices, the Grassmannian manifold of subspaces
of a vector space, and the Kendall manifold of shapes. When the manifold is
known, the full power of the mathematical theory of Riemannian geometry can be
exploitedinboththeformulationofalgorithmsaswellastheirtheoreticalanalysis.
Successful applications of this approach are numerous and range from brain
imaging,kernellearning,andlowrankmatrixcompletion,tocomputervisiontasks
such as object detection and tracking.
This volume focuses on the latter research direction. The forthcoming chapters
were written by researchers currently active in the fields. Overall, the book
describessomeofthelatestalgorithmicadvancesusingRiemanniangeometry,both
v
vi Preface
theoretically and computationally, with potentially large impact on many research
areas in these fields.
The volume targets a broad audience, consisting of Ph.D. students and
researchers in machine learning, statistics, optimization, computer vision, and
related fields.
Acknowledgments
We wish to thank all the authors for contributing some of their latest works to the
volume. We also wish to thank Pierre-Antoine Absil, Gregory Chirikjian, Mark
Girolami, Pavan Turaga, Bart Vandereycken, and Baba Vemuri, for their help in
reviewing the manuscript. Finally, we wish to thank Simon Rees and the Springer
editing team for helping us bring this volume to fruition.
Genoa, Italy Hà Quang Minh
June 2016 Vittorio Murino
Contents
1 Bayesian Statistical Shape Analysis on the Manifold of
Diffeomorphisms .... .... ..... .... .... .... .... .... ..... .... 1
Miaomiao Zhang and P. Thomas Fletcher
2 Sampling Constrained Probability Distributions Using Spherical
Augmentation .. .... .... ..... .... .... .... .... .... ..... .... 25
Shiwei Lan and Babak Shahbaba
3 Geometric Optimization in Machine Learning . .... .... ..... .... 73
Suvrit Sra and Reshad Hosseini
4 Positive Definite Matrices: Data Representation and Applications
to Computer Vision.. .... ..... .... .... .... .... .... ..... .... 93
Anoop Cherian and Suvrit Sra
5 From Covariance Matrices to Covariance Operators: Data
Representation from Finite to Infinite-Dimensional Settings ... .... 115
Hà Quang Minh and Vittorio Murino
6 Dictionary Learning on Grassmann Manifolds .... .... ..... .... 145
Mehrtash Harandi, Richard Hartley, Mathieu Salzmann
and Jochen Trumpf
7 Regression on Lie Groups and Its Application to Affine Motion
Tracking... .... .... .... ..... .... .... .... .... .... ..... .... 173
Fatih Porikli
8 An Elastic Riemannian Framework for Shape Analysis of Curves
and Tree-Like Structures . ..... .... .... .... .... .... ..... .... 187
Adam Duncan, Zhengwu Zhang and Anuj Srivastava
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 207
vii
Contributors
Anoop Cherian ARC Centre of Excellence for Robotic Vision, Australian
National University, Canberra, Australia
AdamDuncan DepartmentofStatistics,FloridaStateUniversity,Tallahassee,FL,
USA
P. Thomas Fletcher University of Utah, Salt Lake City, UT, USA
Mehrtash Harandi College of Engineering and Computer Science, Australian
National University, Canberra, ACT, Australia
Richard Hartley College of Engineering and Computer Science, Australian
National University, Canberra, ACT, Australia
Reshad Hosseini School of ECE, College of Engineering, University of Tehran,
Tehran, Iran
Shiwei Lan Department of Statistics, University of Warwick, Coventry, UK
HàQuangMinh PatternAnalysisandComputerVision(PAVIS),IstitutoItaliano
di Tecnologia (IIT), Genoa, Italy
VittorioMurino PatternAnalysisandComputerVision(PAVIS),IstitutoItaliano
di Tecnologia (IIT), Genoa, Italy
FatihPorikli AustralianNationalUniveristy,Canberra,Australia;Data61/CSIRO,
Eveleigh, Australia
Mathieu Salzmann CVLab, EPFL, Lausanne, Switzerland
BabakShahbaba DepartmentofStatisticsandDepartment ofComputerScience,
University of California, Irvine, CA, USA
Suvrit Sra Laboratory for Information & Decision Systems (LIDS),
Massachusetts Institute of Technology, Cambridge, MA, USA
ix
x Contributors
Anuj Srivastava Department of Statistics, Florida State University, Tallahassee,
FL, USA
Jochen Trumpf College of Engineering and Computer Science, Australian
National University, Canberra, ACT, Australia
Miaomiao Zhang Massachusetts Institute of Technology, Cambridge, MA, USA
Zhengwu Zhang SAMSI, Research Triangle Park, Durham, NC, USA