Exact and Stable Recovery of Rotations for Robust Synchronization (1211.2441v4)

Abstract: The synchronization problem over the special orthogonal group $SO(d)$ consists of estimating a set of unknown rotations $R_1,R_2,...,R_n$ from noisy measurements of a subset of their pairwise ratios $R_{i}^{-1}R_{j}$. The problem has found applications in computer vision, computer graphics, and sensor network localization, among others. Its least squares solution can be approximated by either spectral relaxation or semidefinite programming followed by a rounding procedure, analogous to the approximation algorithms of \textsc{Max-Cut}. The contribution of this paper is three-fold: First, we introduce a robust penalty function involving the sum of unsquared deviations and derive a relaxation that leads to a convex optimization problem; Second, we apply the alternating direction method to minimize the penalty function; Finally, under a specific model of the measurement noise and for both complete and random measurement graphs, we prove that the rotations are exactly and stably recovered, exhibiting a phase transition behavior in terms of the proportion of noisy measurements. Numerical simulations confirm the phase transition behavior for our method as well as its improved accuracy compared to existing methods.

• The paper introduces a novel LUD cost function that improves robustness against outliers in rotation synchronization.
• It reformulates the LUD problem via semidefinite relaxation, ensuring computational tractability with proven exact and stable recovery theorems.
• Numerical simulations show a phase transition in performance, with the method outperforming traditional least squares techniques in noise resilience.

Exact and Stable Recovery of Rotations for Robust Synchronization

The paper "Exact and Stable Recovery of Rotations for Robust Synchronization" presents a novel approach to addressing the synchronization of rotations problem over the special orthogonal group, $SO(d)$. This problem involves estimating a set of unknown rotations from noisy measurements of their pairwise ratios. The paper introduces a robust optimization framework that improves the recovery accuracy of the unknown rotations, particularly in the presence of noise and outliers.

Problem and Approach

The synchronization problem under consideration is pivotal in fields such as computer vision and sensor network localization. Conventional methods for estimating the rotations from noisy measurements involve solving a least squares problem, which can be done through spectral relaxation or semidefinite programming (SDP). While these methods are effective, they are sensitive to outliers, which can lead to significant errors in estimation.

The authors propose an alternative robust estimation method based on the Least Unsquared Deviation (LUD) cost function. This approach minimizes the sum of unsquared residuals, offering robustness against outliers. The LUD method is reformulated as a convex optimization problem through semidefinite relaxation, making it computationally feasible.

Contributions and Theoretical Results

The main contributions of the paper are outlined as follows:

1. Introduction of LUD Function: The paper introduces a new penalty function to improve robustness against outliers by focusing on unsquared deviations, which is shown to yield better results than the traditional squared deviations in the presence of outliers.
2. Semidefinite Relaxation: The authors derive a semidefinite relaxation of the LUD optimization problem, which ensures the problem remains tractable and can be solved efficiently using existing optimization techniques, particularly the alternating direction method.
3. Exact and Stable Recovery Theorems: The paper provides theoretical guarantees for the exact and stable recovery of rotations under specific noise models. The authors show the existence of a critical probability threshold above which the recovery becomes exact. Furthermore, they prove that the recovery is stable to small perturbations in the noise model.
4. Numerical Simulations: Simulations corroborate the theoretical findings, demonstrating a phase transition in the recovery performance and showing that LUD-based recovery significantly outperforms existing methods in terms of accuracy.

Implications and Speculations on Future Developments

The implications of this research are substantial, offering a more reliable method for dealing with synchronization problems in noisy environments. The robust framework provided by the LUD method could be beneficial across various applications that rely on precise angular measurements.

Looking forward, the approach could be extended to more complex measurement graphs and integrated with other probabilistic models to further enhance robustness. Future explorations could also involve adaptive algorithms that dynamically adjust to changing noise levels and optimize performance across diverse real-world settings.

Overall, the paper contributes a significant advancement in robust rotation synchronization, offering insights and tools that could be valuable to both theoretical research and practical applications in numerous technology-driven industries.