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Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces
Minh Ha Quang · Marco San Biagio · Vittorio Murino

This paper introduces a novel mathematical and computational framework, namely {\it Log-Hilbert-Schmidt metric} between positive definite operators on a Hilbert space. This is a generalization of the Log-Euclidean metric on the Riemannian manifold of positive definite matrices to the infinite-dimensional setting. The general framework is applied in particular to compute distances between covariance operators on a Reproducing Kernel Hilbert Space (RKHS), for which we obtain explicit formulas via the corresponding Gram matrices. Empirically, we apply our formulation to the task of multi-category image classification, where each image is represented by an infinite-dimensional RKHS covariance operator. On several challenging datasets, our method significantly outperforms approaches based on covariance matrices computed directly on the original input features, including those using the Log-Euclidean metric, Stein and Jeffreys divergences, achieving new state of the art results.

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Minh Ha Quang (Istituto Italiano di Tecnologia)
Marco San Biagio (IIT - ISTITUTO ITALIANO DI TECNOLOGIA)
Vittorio Murino (Istituto Italiano di Tecnologia)

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