Codebook Training for Trellis-Based Hierarchical Grassmannian Classification

We consider classification of points on a complex-valued Grassmann manifold of
${m}$
-dimensional subspaces within the
${n}$
-dimensional complex Euclidean space. We introduce a trellis-based hierarchical classification network, which is based on an orthogonal product decomposition of the orthogonal basis representing the
${m}$
-dimensional subspace. Exploiting the similarity of the proposed trellis classifier with a neural network, we propose stochastic gradient-based training techniques. We apply the proposed methods to two important applications in wireless communication, namely Grassmannian channel state information quantization in multiple-input multiple-output communications and non-coherent Grassmannian multi-resolution transmission.