Consensus Optimization on Manifolds

TL;DR: The present paper considers distributed consensus algorithms that involve $N$ agents evolving on a connected compact homogeneous manifold, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds.

Abstract: The present paper considers distributed consensus algorithms that involve $N$ agents evolving on a connected compact homogeneous manifold. The agents track no external reference and communicate their relative state according to a communication graph. The consensus problem is formulated in terms of the extrema of a cost function. This leads to efficient gradient algorithms to synchronize (i.e., maximizing the consensus) or balance (i.e., minimizing the consensus) the agents; a convenient adaptation of the gradient algorithms is used when the communication graph is directed and time-varying. The cost function is linked to a specific centroid definition on manifolds, introduced here as the induced arithmetic mean, that is easily computable in closed form and may be of independent interest for a number of manifolds. The special orthogonal group $SO(n)$ and the Grassmann manifold $\text{{\it Grass\/}}(p,n)$ are treated as original examples. A link is also drawn with the many existing results on the circle.