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Covariances in computer vision and machine learning

Author: Quang Minh Hà; Vittorio Murino
Publisher: [San Rafael, California] : Morgan & Claypool, 2018.
Series: Synthesis lectures on computer vision, # 13.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
Covariance matrices play important roles in many areas of mathematics, statistics, and machine learning, as well as their applications. In computer vision and image processing, they give rise to a powerful data representation, namely the covariance descriptor, with numerous practical applications. In this book, we begin by presenting an overview of the finite-dimensional covariance matrix representation approach of  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Quang Minh Hà; Vittorio Murino
ISBN: 9781681730141 1681730146
OCLC Number: 1012748003
Description: 1 online resource (xiii, 156 pages) : illustrations.
Contents: Part I. Covariance matrices and applications --
1. Data representation by covariance matrices --
1.1 Covariance matrices for data representation --
1.2 Statistical interpretation --
2. Geometry of SPD matrices --
2.1 Euclidean distance --
2.2 Interpretations and motivations for the different invariances --
2.3 Basic Riemannian geometry --
2.4 Affine-invariant Riemannian metric on SPD matrices --
2.4.1 Connection with the Fisher-Rao metric --
2.5 Log-Euclidean metric --
2.5.1 Log-Euclidean distance as an approximation of the affine-invariant Riemannian distance --
2.5.2 Log-Euclidean distance as a Riemannian distance --
2.5.3 Log-Euclidean vs. Euclidean --
2.6 Bregman divergences --
2.6.1 Log-determinant divergences --
2.6.2 Connection with the Rényi and Kullback-Leibler divergences --
2.7 Alpha-Beta Log-Det divergences --
2.8 Power Euclidean metrics --
2.9 Distances and divergences between empirical covariance matrices --
2.10 Running time comparison --
2.11 Summary --
3. Kernel methods on covariance matrices --
3.1 Positive definite kernels and reproducing kernel Hilbert spaces --
3.2 Positive definite kernels on SPD matrices --
3.2.1 Positive definite kernels with the Euclidean metric --
3.2.2 Positive definite kernels with the log-Euclidean metric --
3.2.3 Positive definite kernels with the symmetric Stein divergence --
3.2.4 Positive definite kernels with the affine-invariant Riemannian metric --
3.3 Kernel methods on covariance matrices --
3.4 Experiments on image classification --
3.4.1 Datasets --
3.4.2 Results --
3.5 Related approaches --
Part II. Covariance operators and applications --
4. Data representation by covariance operators --
4.1 Positive definite kernels and feature maps --
4.2 Covariance operators in RKHS --
4.3 Data representation by RKHS covariance operators --
5. Geometry of covariance operators --
5.1 Hilbert-Schmidt distance --
5.2 Riemannian distances between covariance operators --
5.2.1 The affine-invariant Riemannian metric --
5.2.2 Log-Hilbert-Schmidt metric --
5.3 Infinite-dimensional alpha log-determinant divergences --
5.4 Summary --
6. Kernel methods on covariance operators --
6.1 Positive definite kernels on covariance operators --
6.1.1 Kernels defined using the Hilbert-Schmidt metric --
6.1.2 Kernels defined using the log-Hilbert-Schmidt metric --
6.2 Two-layer kernel machines --
6.3 Approximate methods --
6.3.1 Approximate log-Hilbert-Schmidt distance and approximate affine-invariant Riemannian distance --
6.3.2 Computational complexity --
6.3.3 Approximate log-Hilbert-Schmidt inner product --
6.3.4 Two-layer kernel machine with the approximate log-Hilbert-Schmidt distance --
6.3.5 Case study: approximation by Fourier feature maps --
6.4 Experiments in image classification --
6.5 Summary --
7. Conclusion and future outlook --
A. Supplementary technical information --
Mean squared errors for empirical covariance matrices --
Matrix exponential and principal logarithm Fréchet derivative --
The quasi-random Fourier features --
Low-discrepancy sequences --
The Gaussian case --
Proofs of several mathematical results --
Bibliography --
Authors' biographies.
Series Title: Synthesis lectures on computer vision, # 13.
Responsibility: Hà Quang Minh, Vittorio Murino.
More information:

Abstract:

Presents an overview of the {\it finite-dimensional covariance matrix} representation approach of images, along with its statistical interpretation. In particular, the book discusses the various  Read more...

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