Robust Soft LVQ With LEML for SPD Matrices

Many learning scenarios involve non-Euclidean data. For instance, in electroencephalogram (EEG) or image classification, the input can be characterized by symmetric positive-definite (SPD) matrices. These matrices live on the curved Riemannian manifold instead of the flat Euclidean space. In such situations, classical Euclidean learning methods may fail due to the non-Euclidean nature of the data. This article proposes to generalize robust soft learning vector quantization (RSLVQ) targeted for Euclidean data to cope with such data in the log-Euclidean framework. Log-Euclidean metric learning (LEML) is incorporated into the RSLVQ framework, jointly learning the prototypes on the manifold and the distance metric tensor of the tangent map that projects the original tangent space to a more discriminative one. Two methods are subsequently proposed to learn the distance metric tensor. It is firstly confined to have full rank and treated as an SPD matrix, which is learned using the log-Euclidean framework. Then, this constraint is removed letting it become a symmetric positive semidefinite (SPSD) matrix, which is learned using the quotient geometry. In addition, we propose to adapt the variance parameter introduced in the probabilistic modelling during the training course of the classifier by the minimization of the negative log likelihood function with respect to this parameter. Experiments on multiple data sets with different properties show the proposed methods have good classification performances and low computational complexities.