We introduce the problem of asymptotic subspace consensus, which requires the outputs of processes to converge onto a common subspace while remaining inside the convex hull of initial vectors.This is a relaxation of asymptotic consensus in which outputs have to converge to a single point, i.e., a zero-dimensional affine subspace. We give a complete characterization of the solvability of asymptotic subspace consensus in oblivious message adversaries. In particular, we show that a large class of algorithms used for asymptotic consensus gracefully degrades to asymptotic subspace consensus in distributed systems with weaker assumptions on the communication network. We also present bounds on the rate by which a lower-than-initial dimension is reached.

翻译：本文提出了渐近子空间共识问题，该问题要求各进程的输出收敛至一个公共子空间，同时保持在初始向量凸包内部。这是对渐近共识问题的松弛化——在渐近共识中，输出必须收敛至单一点（即零维仿射子空间）。我们针对遗忘型消息对抗模型下的渐近子空间共识可解性给出了完整刻画。特别地，我们证明在通信网络假设较弱的分佈式系统中，用于实现渐近共识的广泛算法类能够优雅地降级为渐近子空间共识。本文还给出了达成低于初始维度子空间的收敛速率界限。