Network Linear Equations with Finite Data Rates

In this paper, we propose a distributed quantized algorithm for solving the network linear equation mathbf{z}=mathbf{Hy} subject to digital node communications, where each node only knows a single row of the partitioned matrix [H z]. Each node holds a dynamic state and interacts with its neighbors through an undirected connected graph. Due to the data-rate constraint, each node builds an encoder-decoder pair, with which it produces transmitted message with a zooming-in finite-level uniform quantizer and also generates estimates of its neighbors' states from the received signals. When the equation admits a unique solution, the algorithm drives all nodes' estimates to converge exponentially fast to that solution. When a unique least-squares solution exists, such a solution can be obtained with a suitably selected time-varying step size. In both cases, a minimal data rate with the three-level quantizers is shown to be enough for guaranteeing the desired convergence.