Efficient Detection of Long Consistent Cycles and its Application to Distributed Synchronization

TL;DR

This paper introduces an efficient algorithm for detecting long consistent cycles in group synchronization, significantly improving computational performance and enabling practical distributed synchronization applications.

Contribution

The paper presents a novel algorithm leveraging cycles of length 3 to 6 with improved complexity, and provides theoretical analysis and practical validation for distributed synchronization.

Findings

Achieves $O(n^3)$ complexity, or $O(n^{2.373})$ with faster algorithms.

Demonstrates state-of-the-art performance in distributed synchronization.

Provides theoretical guarantees under the uniform corruption model.

Abstract

Group synchronization plays a crucial role in global pipelines for Structure from Motion (SfM). Its formulation is nonconvex and it is faced with highly corrupted measurements. Cycle consistency has been effective in addressing these challenges. However, computationally efficient solutions are needed for cycles longer than three, especially in practical scenarios where 3-cycles are unavailable. To overcome this computational bottleneck, we propose an algorithm for group synchronization that leverages information from cycles of lengths ranging from three to six with a time complexity of order $O(n^3)$ (or $O(n^{2.373})$ when using a faster matrix multiplication algorithm). We establish non-trivial theory for this and related methods that achieves competitive sample complexity, assuming the uniform corruption model. To advocate the practical need for our method, we consider distributed group synchronization, which requires at least 4-cycles, and we illustrate state-of-the-art performance by our method in this context.