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This paper presents a novel framework for visual object recognition usinginfinite-dimensional covariance operators of input features in the paradigm ofkernel methods on infinite-dimensional Riemannian manifolds. Our formulationprovides in particular a rich representation of image features by exploitingtheir non-linear correlations. Theoretically, we provide a finite-dimensionalapproximation of the Log-Hilbert-Schmidt (Log-HS) distance between covarianceoperators that is scalable to large datasets, while maintaining an effectivediscriminating capability. This allows us to efficiently approximate anycontinuous shift-invariant kernel defined using the Log-HS distance. At thesame time, we prove that the Log-HS inner product between covariance operatorsis only approximable by its finite-dimensional counterpart in a very limitedscenario. Consequently, kernels defined using the Log-HS inner product, such aspolynomial kernels, are not scalable in the same way as shift-invariantkernels. Computationally, we apply the approximate Log-HS distance formulationto covariance operators of both handcrafted and convolutional features,exploiting both the expressiveness of these features and the power of thecovariance representation. Empirically, we tested our framework on the task ofimage classification on twelve challenging datasets. In almost all cases, theresults obtained outperform other state of the art methods, demonstrating thecompetitiveness and potential of our framework.

Kernel Methods on Approximate Infinite-Dimensional Covariance Operators for Image Classification