Low-Rank Matrix Completion for Topological Interference Management by Riemannian Pursuit

In this paper, we present a flexible low-rank matrix completion (LRMC) approach for topological interference management (TIM) in the partially connected $K$-user interference channel. No channel state information (CSI) is required at the transmitters except the network topology information. The previous attempt on the TIM problem is mainly based on its equivalence to the index coding problem, but so far only a few index coding problems have been solved. In contrast, in this paper, we present an algorithmic approach to investigate the achievable degrees-of-freedom (DoFs) by recasting the TIM problem as an LRMC problem. Unfortunately, the resulting LRMC problem is known to be NP-hard, and the main contribution of this paper is to propose a Riemannian pursuit (RP) framework to detect the rank of the matrix to be recovered by iteratively increasing the rank. This algorithm solves a sequence of fixed-rank matrix completion problems. To address the convergence issues in the existing fixed-rank optimization methods, the quotient manifold geometry of the search space of fixed-rank matrices is exploited via Riemannian optimization. By further exploiting the structure of the low-rank matrix varieties, i.e., the closure of the set of fixed-rank matrices, we develop an efficient rank increasing strategy to find good initial points in the procedure of rank pursuit. Simulation results demonstrate that the proposed RP algorithm achieves a faster convergence rate and higher achievable DoFs for the TIM problem compared with the state-of-the-art methods.