  Large-scale distributed systems such as sensor networks, often need to
achieve filtering and consensus on an estimated parameter from high-dimensional
measurements. Running a Kalman filter on every node in such a network is
computationally intensive; in particular the matrix inversion in the Kalman
gain update step is expensive. In this paper, we extend previous results in
distributed Kalman filtering and large-scale machine learning to propose a
gradient descent step for updating an estimate of the error covariance matrix;
this is then embedded and analyzed in the context of distributed Kalman
filtering. We provide properties of the resulting filters, in addition to a
number of applications throughout the paper.
