  This paper presents a novel framework for visual object recognition using
infinite-dimensional covariance operators of input features in the paradigm of
kernel methods on infinite-dimensional Riemannian manifolds. Our formulation
provides in particular a rich representation of image features by exploiting
their non-linear correlations. Theoretically, we provide a finite-dimensional
approximation of the Log-Hilbert-Schmidt (Log-HS) distance between covariance
operators that is scalable to large datasets, while maintaining an effective
discriminating capability. This allows us to efficiently approximate any
continuous shift-invariant kernel defined using the Log-HS distance. At the
same time, we prove that the Log-HS inner product between covariance operators
is only approximable by its finite-dimensional counterpart in a very limited
scenario. Consequently, kernels defined using the Log-HS inner product, such as
polynomial kernels, are not scalable in the same way as shift-invariant
kernels. Computationally, we apply the approximate Log-HS distance formulation
to covariance operators of both handcrafted and convolutional features,
exploiting both the expressiveness of these features and the power of the
covariance representation. Empirically, we tested our framework on the task of
image classification on twelve challenging datasets. In almost all cases, the
results obtained outperform other state of the art methods, demonstrating the
competitiveness and potential of our framework.
