Acknowledgments
Introduction
Covariance Matrices and Applications
Data Representation by Covariance Matrices
Covariance Matrices for Data Representation
Statistical Interpretation
Geometry of SPD Matrices
Euclidean Distance
Interpretations and Motivations for the Different Invariances
Basic Riemannian Geometry
Affine-invariant Riemannian Metric on SPD Matrices
Connection with the Fisher-Rao Metric
Log-Euclidean Metric
Log-Euclidean Distance as an Approximation of the Affine-invariant Riemannian Distance
Log-Euclidean Distance as a Riemannian Distance
Log-Euclidean vs. Euclidean
Bregman Divergences
Log-determinant Divergences
Connection with the RΓ©nyi and Kullback-Leibler Divergences
Alpha-Beta Log-Det Divergences
Power Euclidean Metrics
Distances and Divergences between Empirical Covariance Matrices
Running Time Comparison
Summary
Kernel Methods on Covariance Matrices
Positive Definite Kernels and Reproducing Kernel Hilbert Spaces
Positive Definite Kernels on SPD Matrices
Positive Definite Kernels with the Euclidean Metric
Positive Definite Kernels with the Log-Euclidean Metric
Positive Definite Kernels with the Symmetric Stein Divergence
Positive Definite Kernels with the Affine-invariant Riemannian Metric
Kernel Methods on Covariance Matrices
Experiments on Image Classification
Datasets
Results
Related Approaches
Covariance Operators and Applications
Data Representation by Covariance Operators
Positive Definite Kernels and Feature Maps
Covariance Operators in RKHS
Data Representation by RKHS Covariance Operators
Geometry of Covariance Operators
Hilbert-Schmidt Distance
Riemannian Distances Between Covariance Operators
The Affine-invariant Riemannian Metric
Log-Hilbert-Schmidt Metric
Infinite-dimensional Alpha Log-determinant Divergences
Summary
Kernel Methods on Covariance Operators
Positive Definite Kernels on Covariance Operators
Kernels Defined Using the Hilbert-Schmidt Metric
Kernels Defined Using the Log-Hilbert-Schmidt Metric
Two-layer Kernel Machines
Approximate Methods
Approximate Log-Hilbert-Schmidt Distance and Approximate Affine-invariant Riemannian Distance
Computational Complexity
Approximate Log-Hilbert-Schmidt Inner Product
Two-Layer Kernel Machine with the Approximate Log-Hilbert-Schmidt Distance
Case Study: Approximation by Fourier Feature Maps
Experiments in Image Classification
Summary
Conclusion and Future Outlook
Supplementary Technical Information
Mean Squared Errors for Empirical Covariance Matrices
Matrix Exponential and Principal Logarithm
FrΓ©chet Derivative
The Quasi-random Fourier Features
Low-discrepancy Sequences
The Gaussian Case
Proofs of Several Mathematical Results
Bibliography
Authors' Biographies
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