# A Lifting method for analyzing distributed synchronization on the unit sphere

arXiv: https://arxiv.org/abs/1805.02528 · HTML: https://mlsanity.com/paper/1805.02528/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2018.

## Built on by (most cited first, 4 of 4)

- [The Asymptotic Behavior of Attention in Transformers](https://mlsanity.com/paper/2412.02682/) (2024)
- [SyncDiff: Synchronized Motion Diffusion for Multi-Body Human-Object Interaction Synthesis](https://mlsanity.com/paper/2412.20104/) (2024)
- [Multistability of Self-Attention Dynamics in Transformers](https://mlsanity.com/paper/2511.11553/) (2025)
- [Self-Attention Dynamics with Rotary Position Embeddings: Twisted States and Explicit Consensus Rates on the Sphere](https://mlsanity.com/paper/2607.24502/) (2026)

## Abstract (arXiv)

This paper introduces a new lifting method for analyzing convergence of continuous-time distributed synchronization/consensus systems on the unit sphere. Points on the d-dimensional unit sphere are lifted to the (d+1)-dimensional Euclidean space. The consensus protocol on the unit sphere is the classical one, where agents move toward weighted averages of their neighbors in their respective tangent planes. Only local and relative state information is used. The directed interaction graph topologies are allowed to switch as a function of time. The dynamics of the lifted variables are governed by a nonlinear consensus protocol for which the weights contain ratios of the norms of state variables. We generalize previous convergence results for hemispheres. For a large class of consensus protocols defined for switching uniformly quasi-strongly connected time-varying graphs, we show that the consensus manifold is uniformly asymptotically stable relative to closed balls contained in a hemisphere. Compared to earlier projection based approaches used in this context such as the gnomonic projection, which is defined for hemispheres only, the lifting method applies globally. With that, the hope is that this method can be useful for future investigations on global convergence.