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

Shaohan Li, Yunpeng Shi, Gilad Lerman

TL;DR

This work tackles rotation synchronization in global Structure from Motion under heavy corruption by extending cycle-consistency methods beyond 3-cycles. It introduces LongSync, a scalable, vectorized variant of Cycle Edge Message Passing (CEMP) that leverages long cycles with the chordal distance $\mathcal{D}$ and weighted quadratic averaging to avoid enumerating all cycles, achieving per-iteration time $O(n^3)$ for cycle lengths up to $c\le6$. The authors provide formal recovery guarantees under the Uniform Corruption Model with near-optimal sample complexity and demonstrate substantial practical gains in distributed SfM, including a real-data evaluation on the PhotoTourism dataset. The approach generalizes to linear groups beyond $SO(3)$ and shows promise for large-scale, robust, distributed synchronization tasks in computer vision and related domains.

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.

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

TL;DR

This work tackles rotation synchronization in global Structure from Motion under heavy corruption by extending cycle-consistency methods beyond 3-cycles. It introduces LongSync, a scalable, vectorized variant of Cycle Edge Message Passing (CEMP) that leverages long cycles with the chordal distance and weighted quadratic averaging to avoid enumerating all cycles, achieving per-iteration time for cycle lengths up to . The authors provide formal recovery guarantees under the Uniform Corruption Model with near-optimal sample complexity and demonstrate substantial practical gains in distributed SfM, including a real-data evaluation on the PhotoTourism dataset. The approach generalizes to linear groups beyond and shows promise for large-scale, robust, distributed synchronization tasks in computer vision and related domains.

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 (or 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.
Paper Structure (23 sections, 14 theorems, 66 equations, 3 figures, 4 tables, 2 algorithms)

This paper contains 23 sections, 14 theorems, 66 equations, 3 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions of This Work
  4. Problem Formulation and Preliminaries
  5. Notations and Definitions
  6. Review of CEMP for $c$-Cycles
  7. Our method: LongSync
  8. LongSync: Modification of CEMP
  9. Computational Complexity
  10. Theory for Uniform Corruption Model
  11. Synthetic Data Experiment
  12. Uniform Corruption Model
  13. Uniform Bipartite Corruption Model
  14. Real Data Experiment
  15. Conclusion
  16. ...and 8 more sections

Key Result

Proposition 3.1

The update rule of LongSync eqn:sij_linear is equivalent to the following matrix equations: where $\boldsymbol{W}^{(t+1)} = \boldsymbol{A} \odot \exp(-\beta_t \boldsymbol{S}^{(t)})$ and Here $\exp$ denotes the elementwise exponential function.

Figures (3)

  • Figure 1: Average errors for IRLS, CEMP+IRLS and LongSync +IRLS with $c=4$, 5, using the uniform corruption (top) and uniform bipartite corruption (bottom) models. The mean errors are measured in degrees. LongSync4 and LongSync5 refer to LongSync with 4 and 5 cycles, respectively.
  • Figure 2: Median error for rotations for each dataset measured in degrees.
  • Figure 3: Runtime ratio and average median and mean error gaps between the distributed methods and MPLS on the entire graph.

Theorems & Definitions (27)

  • Proposition 3.1
  • Theorem 4.1
  • Theorem C.1
  • proof
  • Theorem C.2
  • Remark C.3
  • Definition C.4
  • Theorem C.5
  • Theorem C.6
  • Lemma C.7
  • ...and 17 more