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 {\it finite-dimensional
covariance matrix} representation approach of images, along with
its statistical interpretation. In particular, we discuss the
various distances and divergences that arise from the intrinsic
geometrical structures of the set of Symmetric Positive Definite
(SPD) matrices, namely Riemannian manifold and convex cone
structures. Computationally, we focus on kernel methods on
covariance matrices, especially using the Log-Euclidean distance.
We then show some of the latest developments in the generalization
of the finite-dimensional covariance matrix representation to the
{\it infinite-dimensional covariance operator} representation via
positive definite kernels. We present the generalization of the
affine-invariant Riemannian metric and the Log-Hilbert-Schmidt
metric, which generalizes the Log-Euclidean distance.
Computationally, we focus on kernel methods on covariance
operators, especially using the Log-Hilbert-Schmidt distance.
Specifically, we present a two-layer kernel machine, using the
Log-Hilbert-Schmidt distance and its finite-dimensional
approximation, which reduces the computational complexity of the
exact formulation while largely preserving its capability.
Theoretical analysis shows that, mathematically, the approximate
Log-Hilbert-Schmidt distance should be preferred over the
approximate Log-Hilbert-Schmidt inner product and, computationally,
it should be preferred over the approximate affine-invariant
Riemannian distance. Numerical experiments on image classification
demonstrate significant improvements of the infinite-dimensional
formulation over the finite-dimensional counterpart. Given the
numerous applications of covariance matrices in many areas of
mathematics, statistics, and machine learning, just to name a few,
we expect that the infinite-dimensional covariance operator
formulation presented here will have many more applications beyond
those in computer vision.
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