  Symmetric Positive Definite (SPD) matrix learning methods have become popular
in many image and video processing tasks, thanks to their ability to learn
appropriate statistical representations while respecting Riemannian geometry of
underlying SPD manifolds. In this paper we build a Riemannian network
architecture to open up a new direction of SPD matrix non-linear learning in a
deep model. In particular, we devise bilinear mapping layers to transform input
SPD matrices to more desirable SPD matrices, exploit eigenvalue rectification
layers to apply a non-linear activation function to the new SPD matrices, and
design an eigenvalue logarithm layer to perform Riemannian computing on the
resulting SPD matrices for regular output layers. For training the proposed
deep network, we exploit a new backpropagation with a variant of stochastic
gradient descent on Stiefel manifolds to update the structured connection
weights and the involved SPD matrix data. We show through experiments that the
proposed SPD matrix network can be simply trained and outperform existing SPD
matrix learning and state-of-the-art methods in three typical visual
classification tasks.
