Covariances in Computer Vision and Machine Learning

Covariances in Computer Vision and Machine Learning

Ha Quang Minh, Vittorio Murino
ISBN: 9781681730134 | PDF ISBN: 9781681730141
Hardcover ISBN:9781681732596
Copyright © 2017 | 165 Pages | Publication Date: October, 2017

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This book presents an overview of the finite-dimensional covariance matrix data representations and shows some of the latest developments for their generalization to the infinite-dimensional covariance operator representation via positive definite kernels. Along with the statistical interpretation of the different approaches, it presents several practical applications using machine learning methods for covariance matrices and covariance operators, especially in the domain of computer vision. In particular, numerical experiments on image classification demonstrate the significant improvements of the infinite-dimensional formulation over the finite-dimensional counterpart.

Given the important role played by covariance matrices in many areas of mathematics, statistics, and machine learning, to name a few, the authors expect that the infinite-dimensional covariance operator formulation presented here will have many more applications beyond those in computer vision.

Table of Contents

Introduction

Part I: Covariance Matrices and Applications
Data Representation by Covariance Matrices
Geometry of SPD matrices
Kernel Methods on Covariance Matrices

Part II: Covariance Operators and Applications
Data Representation by Covariance Operators
Geometry of Covariance Operators
Kernel Methods on Covariance Operators

Appendix: Supplementary Technical Information


About the Author(s)

Ha Quang Minh, Istituto Italiano di Tecnologia
Ha Quang Minh received the Ph.D. degree in mathematics from Brown University, Providence, RI, USA, in May 2006, under the supervision of Steve Smale. He is currently a Researcher in the Department of Pattern Analysis and Computer Vision (PAVIS) with the Istituto Italiano di Tecnologia (IIT), Genova, Italy. Prior to joining IIT, he held research positions at the University of Chicago, the University of Vienna, Austria, and Humboldt University of Berlin, Germany. He was also a Junior Research Fellow at the Erwin Schrodinger International Institute for Mathematical Physics in Vienna and a Fellow at the Institute for Pure and Applied Mathematics (IPAM) at the University of California, Los Angeles (UCLA). His current research interests include applied and computational functional analysis, applied and computational di erential geometry, machine learning, computer vision, and image and signal processing. His recent research contributions include the infinite-dimensional Log-Hilbert-Schmidt metric and Log-Determinant divergences between positive definite operators, along with their applications in machine learning and computer vision in the setting of kernel methods. He received the Microsoft Best Paper Award at the Conference on Uncertainty in Artificial Intelligence (UAI) in 2013 and the IBM Pat Goldberg Memorial Best Paper Award in Computer Science, Electrical Engineering, and Mathematics in 2013.

Vittorio Murino, Istituto Italiano di Tecnologia
Vittorio Murino is full professor and head of the Pattern Analysis and Computer Vision (PAVIS) department at the Istituto Italiano di Tecnologia (IIT), Genoa, Italy. He received the Ph.D. in Electronic Engineering and Computer Science in 1993 at the University of Genoa, Italy. Then, he was first at the University of Udine and, since 1998, at the University of Verona, where he was chairman of the Department of Computer Science from 2001 to 2007. His research interests are in computer vision and machine learning, in particular, probabilistic techniques for image and video analysis with applications to video surveillance, biomedical image analysis and bio-informatics. He is currently a member of the editorial board of Computer Vision and Image Understanding, Pattern Analysis and Applications, and Machine Vision & Applications journals. He was also associate editor of Pattern Recognition and of the IEEE Transactions on Systems, Man, and Cybernetics until 2016. Finally, he is a Senior Member of the IEEE since 2002 and IAPR Fellow since 2006.

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