Covariances in Computer Vision and Machine Learning

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 Rényi 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
		Fréchet Derivative
		The Quasi-random Fourier Features
			Low-discrepancy Sequences
			The Gaussian Case
		Proofs of Several Mathematical Results
	Bibliography
	Authors\' Biographies
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