How to Understand LMMSE Transceiver Design for MIMO Systems From Quadratic Matrix Programming

In this paper, a unified linear minimum mean-square-error (LMMSE) transceiver design framework is investigated, which is suitable for a wide range of wireless systems. The unified design is based on an elegant and powerful mathematical programming technology termed as quadratic matrix programming (QMP). Based on QMP it can be observed that for different wireless systems, there are certain common characteristics which can be exploited to design LMMSE transceivers e.g., the quadratic forms. It is also discovered that evolving from a point-to-point MIMO system to various advanced wireless systems such as multi-cell coordinated systems, multi-user MIMO systems, MIMO cognitive radio systems, amplify-and-forward MIMO relaying systems and so on, the quadratic nature is always kept and the LMMSE transceiver designs can always be carried out via iteratively solving a number of QMP problems. A comprehensive framework on how to solve QMP problems is also given. The work presented in this paper is likely to be the first shoot for the transceiver design for the future ever-changing wireless systems.Comment: 31 pages, 4 figures, Accepted by IET Communication

Simplified Energy Landscape for Modularity Using Total Variation

Networks capture pairwise interactions between entities and are frequently used in applications such as social networks, food networks, and protein interaction networks, to name a few. Communities, cohesive groups of nodes, often form in these applications, and identifying them gives insight into the overall organization of the network. One common quality function used to identify community structure is modularity. In Hu et al. [SIAM J. App. Math., 73(6), 2013], it was shown that modularity optimization is equivalent to minimizing a particular nonconvex total variation (TV) based functional over a discrete domain. They solve this problem, assuming the number of communities is known, using a Merriman, Bence, Osher (MBO) scheme. We show that modularity optimization is equivalent to minimizing a convex TV-based functional over a discrete domain, again, assuming the number of communities is known. Furthermore, we show that modularity has no convex relaxation satisfying certain natural conditions. We therefore, find a manageable non-convex approximation using a Ginzburg Landau functional, which provably converges to the correct energy in the limit of a certain parameter. We then derive an MBO algorithm with fewer hand-tuned parameters than in Hu et al. and which is 7 times faster at solving the associated diffusion equation due to the fact that the underlying discretization is unconditionally stable. Our numerical tests include a hyperspectral video whose associated graph has 2.9x10^7 edges, which is roughly 37 times larger than was handled in the paper of Hu et al.Comment: 25 pages, 3 figures, 3 tables, submitted to SIAM J. App. Mat