  We propose a computationally efficient algorithm for the device activity
detection problem in the multi-cell massive multi-input multi-output (MIMO)
system, where the active devices transmit their signature sequences to multiple
BSs in multiple cells and all the BSs cooperate to detect the active devices.
The device activity detection problem has been formulated as a maximum
likelihood maximization (MLE) problem in the literature. The state-of-the-art
algorithm for solving the problem is the (random) coordinate descent (CD)
algorithm. However, the CD algorithm fails to exploit the special sparsity
structure of the solution of the device activity detection problem, i.e., most
of devices are not active in each time slot. In this paper, we propose a novel
active set selection strategy to accelerate the CD algorithm and propose an
efficient active set CD algorithm for solving the considered problem.
Specifically, at each iteration, the proposed active set CD algorithm first
selects a small subset of all devices, namely the active set, which contains a
few devices that contribute the most to the deviation from the first-order
optimality condition of the MLE problem thus potentially can provide the most
improvement to the objective function, then applies the CD algorithm to perform
the detection for the devices in the active set. Simulation results show that
the proposed active set CD algorithm significantly outperforms the
state-of-the-art CD algorithm in terms of the computational efficiency.
