Consensus optimization on manifolds

TL;DR

This paper develops gradient-based consensus algorithms for agents on manifolds, including SO(n) and Grassmann manifolds, optimizing a cost function related to a new centroid concept, with applications to synchronization and balancing.

Contribution

It introduces a novel cost function and a corresponding centroid definition on manifolds, enabling efficient consensus algorithms for agents on complex geometric spaces.

Findings

Algorithms successfully synchronize agents on SO(n) and Grassmann manifolds.

The centroid concept simplifies computations and is applicable to various manifolds.

Directed and time-varying communication graphs are effectively handled.

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 Gr(p,n) are treated as original examples. A link is also drawn with the many existing results on the circle.