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An Interface between Grassmann manifolds and vector spaces

In this paper, we propose a method to map data from a Grassmann manifold to a vector space while maximizing discrimination capability for classification. Subspaces are a practical and robust representation for image set recognition. However, as they exist on a Grassmann manifold, machine learning tools constructed on Euclidean geometry cannot be promptly utilized. Recently, methods to construct end-to-end learnable models for subspaces are starting to be explored, but they require multiple matrix decompositions and can be hard to compute and extend. Therefore we introduce a layer to map Grassmann manifold-valued data to vector space, in such a way that it can be seamlessly used as a layer along with other powerful tools defined on Euclidean space. The key idea of our method is to formulate the manifold logarithmic map (log) as a learnable model, where we seek to learn a tangency point that minimizes a loss function with respect to the data. The log effectively transforms a manifold point into a tangent vector. This log model can be learned with Riemannian stochastic gradient descent on the target manifold. We demonstrate the effectiveness of our proposed method on the applications of hand shape recognition, face identification and facial emotion recognition.