$k$-Dimensional Agreement in Multiagent Systems

Given a network of agents, we study the problem of designing a distributed algorithm that computes $k$ independent weighted means of the network's initial conditions (namely, the agents agree on a $k$-dimensional space). Akin to average consensus, this problem finds applications in distributed computing and sensing, where agents seek to simultaneously evaluate $k$ independent functions at a common point by running a single coordination algorithm. We show that linear algorithms can agree on quantities that are oblique projections of the vector of initial conditions, and we provide techniques to design protocols that are compatible with a pre-specified communication graph. More broadly, our results show that a single agreement algorithm can solve $k$ consensus problems simultaneously at a fraction of the complexity of classical approaches but, in general, it requires higher network connectivity.