Codebook Training for Trellis-Based Hierarchical Grassmannian Classification

Abstract

We consider classification of points on a complexvalued 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.

Modular CSI Quantization for FDD Massive MIMO Communication

“…where d p ( W, R) is the projection distortion (25), and W is a codebook of N t × K matrices. As we are separately interested in the K columns, Grassmannian codebooks [23], [28], [31], [32] of K-dimensional subspaces are not of interest. Also, due to the modular design in mind, the phases of the K columns are irrelevant.…”

“…The complexity of finding a codeword using (32) is the Kth root of the complexity of finding a codeword in matrix codebook W with the same granularity. Columnwise Grassmann line quantization has the added benefit that it operates natively on the flag manifold, as opposed to a matrix Grassmann [23], [28], [31], [32] or Stiefel [30] quantization. However, if V is overcomplete, there is a high probability that VV H = I Nt .…”

“…Motivated by this, we develop two wideband quantization approaches, namely orthonormalized wideband precoding (OWP) and sequential wideband precoding (SWP), that improve not only the quality of wideband feedback but also the accuracy of subband channel projection-they enable high-granularity feedback of orthogonal bases, with the same complexity as (32).…”

“…In [29], channel Gram matrices are constructed, and information about them is fed back using scalar quantization of matrix elements. Hierarchical quantization of N t × K unitary (Stiefel) matrices as well as K-dimensional subspaces was considered in [30]- [32], where the dimensionality of the subspace decreases when proceeding in the hierarchy. In contrast, [24]- [26] consider independent vector quantization of the columns of U K , i.e.…”

“…Note that OWP of Section V-A1 does not reduce the projection distance as compared to using the possibly nonorthogonal W produced by parallel quantization in (32). OWP only produces effective channels which can be quantized based on isometry.…”