Mind the Gap: Norm-Aware Adaptive Robust Loss for Multivariate Least-Squares Problems (2206.09215v3)

Abstract: Measurement outliers are unavoidable when solving real-world robot state estimation problems. A large family of robust loss functions (RLFs) exists to mitigate the effects of outliers, including newly developed adaptive methods that do not require parameter tuning. All of these methods assume that residuals follow a zero-mean Gaussian-like distribution. However, in multivariate problems the residual is often defined as a norm, and norms follow a Chi-like distribution with a non-zero mode value. This produces a "mode gap" that impacts the convergence rate and accuracy of existing RLFs. The proposed approach, "Adaptive MB," accounts for this gap by first estimating the mode of the residuals using an adaptive Chi-like distribution. Applying an existing adaptive weighting scheme only to residuals greater than the mode leads to more robust performance and faster convergence times in two fundamental state estimation problems, point cloud alignment and pose averaging.

• The paper presents Adaptive MB, which computes a Maxwell-Boltzmann mode estimate to enhance robust residual weighting in state estimation.
• It improves convergence and accuracy in multivariate least-squares problems, notably in point cloud alignment and pose averaging.
• Numerical results show lower median errors and faster execution times compared to traditional robust loss functions in high-outlier scenarios.

Overview of "Mind the Gap: Norm-Aware Adaptive Robust Loss for Multivariate Least-Squares Problems"

This paper addresses a critical challenge in robotics and state estimation: the handling of measurement outliers in multivariate least-squares problems. The authors, Thomas Hitchcox and James Richard Forbes, introduce a novel approach termed "Adaptive MB," which deliberately accounts for the distributional characteristics of residuals in robotic state estimation. This approach is designed to improve the performance of optimization algorithms, particularly in scenarios where robust estimation is crucial, such as point cloud alignment and pose averaging.

Background and Motivation

In state estimation, sensor inaccuracies or incorrect data associations often introduce outliers, leading to significant deviations in state estimation accuracy. Traditional robust loss functions (RLFs), including adaptive methods, typically assume a Gaussian-like distribution with a mode at zero for the residuals. However, in multivariate settings, residuals commonly represented as norms follow a Chi-like distribution with a non-zero mode. This discrepancy, identified as the "mode gap," can adversely affect the convergence and accuracy of existing RLFs.

Key Contributions

The primary contribution of the paper is the introduction of the "Adaptive MB" method, which corrects for the mode gap by refining how residuals are weighted during optimization. The process involves these essential steps:

• Mode Estimation: By fitting a Maxwell-Boltzmann distribution to residuals, the mode is estimated, accounting for their Chi-like distribution.
• Mode-Aware Weighting: Only residuals exceeding this estimated mode are considered outliers, allowing normal inlier residuals to maintain higher weights.
• Extended Adaptability: The method extends on previous adaptive approaches by incorporating this mode-shift into the adaptation process, leading to improved robustness and convergence speeds.

Numerical Results

The method's efficacy is evaluated in two state estimation contexts:

1. Point Cloud Alignment (ICP Framework): Across various datasets representing different environmental complexities, Adaptive MB consistently provided lower median errors and reduced error variability. The convergence speed, measured in iterations and execution time, surpassed current adaptive techniques.
2. Pose Averaging: In high-outlier simulations, Adaptive MB delivered lower error distributions and faster execution times, demonstrating stability across different levels of outlier contamination.

Implications and Future Directions

The Adaptive MB method's consideration of residual distribution characteristics represents a significant step forward for robust loss frameworks in multivariate least-squares problems. It effectively enhances the robustness of state estimation solutions by addressing inherent statistical distributional gaps overlooked by previous models.

Future research could expand on this method to handle more complex state estimation challenges, particularly in dynamic and highly uncertain environments. Furthermore, exploring the integration of this adaptive framework into real-time systems might present additional challenges and opportunities, especially considering computational constraints in time-sensitive applications.

In summary, the Adaptive MB approach provides a compelling framework for enhancing the robustness of robotic state estimation tasks and highlights the importance of accounting for distributional characteristics to mitigate the effects of outliers effectively.