Clustering Complex Subspaces in Large Dimensions

A methodology to cluster multiple sets of Gaussian multivariate complex observations based on the alignment of their column spaces is presented. These subspaces are identified with points in the Grassmann manifold and compared according to a similarity measure drawn from a chosen manifold distance, which is proportional to the squared projection–Frobenius norm. In order to guarantee that distances between subspaces of different dimensions are comparable, we proposed to normalise the corresponding decision statistics with respect to their asymptotic mean and variance, assuming that (i) the dimensions of both the observation and the involved subspaces are large but comparable in magnitude and (ii) both subspaces are generated by the same statistical law. A procedure is derived to estimate these normalisation parameters, leading to a new statistic that can be built exclusively from the observations. The method is applied to a MIMO wireless channel clustering problem, where is shown to outperform conventional similarity measures in terms of classification performance.