# Log-Euclidean Metric Learning on Symmetric Positive Definite Manifold with Application to Image Set Classification

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Zhiwu Huang,
Ruiping Wang,
Shiguang Shan,
Xianqiu Li,
Xilin Chen
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Proceedings of the 32nd International Conference on Machine Learning, PMLR 37:720-729, 2015.

#### Abstract

The manifold of Symmetric Positive Definite (SPD) matrices has been successfully used for data representation in image set classification. By endowing the SPD manifold with Log-Euclidean Metric, existing methods typically work on vector-forms of SPD matrix logarithms. This however not only inevitably distorts the geometrical structure of the space of SPD matrix logarithms but also brings low efficiency especially when the dimensionality of SPD matrix is high. To overcome this limitation, we propose a novel metric learning approach to work directly on logarithms of SPD matrices. Specifically, our method aims to learn a tangent map that can directly transform the matrix logarithms from the original tangent space to a new tangent space of more discriminability. Under the tangent map framework, the novel metric learning can then be formulated as an optimization problem of seeking a Mahalanobis-like matrix, which can take the advantage of traditional metric learning techniques. Extensive evaluations on several image set classification tasks demonstrate the effectiveness of our proposed metric learning method.

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