Outlier-Robust Estimation: Hardness, Minimally Tuned Algorithms, and Applications (2007.15109v3)

Abstract: Nonlinear estimation in robotics and vision is typically plagued with outliers due to wrong data association, or to incorrect detections from signal processing and machine learning methods. This paper introduces two unifying formulations for outlier-robust estimation, Generalized Maximum Consensus (G-MC) and Generalized Truncated Least Squares (G-TLS), and investigates fundamental limits, practical algorithms, and applications. Our first contribution is a proof that outlier-robust estimation is inapproximable: in the worst case, it is impossible to (even approximately) find the set of outliers, even with slower-than-polynomial-time algorithms (particularly, algorithms running in quasi-polynomial time). As a second contribution, we review and extend two general-purpose algorithms. The first, Adaptive Trimming (ADAPT), is combinatorial, and is suitable for G-MC; the second, Graduated Non-Convexity (GNC), is based on homotopy methods, and is suitable for G-TLS. We extend ADAPT and GNC to the case where the user does not have prior knowledge of the inlier-noise statistics (or the statistics may vary over time) and is unable to guess a reasonable threshold to separate inliers from outliers (as the one commonly used in RANSAC). We propose the first minimally tuned algorithms for outlier rejection, that dynamically decide how to separate inliers from outliers. Our third contribution is an evaluation of the proposed algorithms on robot perception problems: mesh registration, image-based object detection (shape alignment), and pose graph optimization. ADAPT and GNC execute in real-time, are deterministic, outperform RANSAC, and are robust up to 80-90% outliers. Their minimally tuned versions also compare favorably with the state of the art, even though they do not rely on a noise bound for the inliers.

• The paper establishes fundamental limits on the approximability of outlier-robust estimation and introduces generalized formulations (G, G-TLS).
• It presents minimally tuned algorithms, ADAPT and GNC, which reduce reliance on user parameters for outlier distinction.
• Experimental results show these algorithms are robust against high outlier rates (up to 90%) and outperform traditional methods like RANSAC in robotics applications.

Overview of "Outlier-Robust Estimation: Hardness, Minimally Tuned Algorithms, and Applications"

The paper "Outlier-Robust Estimation: Hardness, Minimally Tuned Algorithms, and Applications" presents a comprehensive paper on the topic of outlier-robust estimation within the fields of robotics and computer vision. The authors propose two novel formulations, Generalized Maximum Consensus (G) and Generalized Truncated Least Squares (G-TLS), which unify existing approaches to robust estimation and tackle the problem of algorithmic hardness and tractability.

Key Contributions

1. Fundamental Limits of Outlier-Robust Estimation: The paper begins by establishing the inapproximability of outlier-robust estimation in quasi-polynomial time frameworks, subject to well-known complexity assumptions. This reveals a computational barrier in efficiently solving robust estimation problems even when relaxed conditions are adopted.
2. General Formulations: The authors introduce two formulations for robust estimation. The Generalized Maximum Consensus (G) formulation builds upon the maximum consensus problem by enforcing a cumulative residual constraint. On the other hand, the Generalized Truncated Least Squares (G-TLS) serves as an enhanced version of truncated least squares by penalizing outliers more aggressively. These provide a theoretical backbone to explore robust estimation algorithms across various applications.
3. Algorithms with Minimal Tuning: The paper offers minimally tuned variants of two main algorithms, Adaptive Trimming (ADAPT) and Graduated Non-Convexity (GNC). These algorithms are significant in the sense that they reduce the dependency on specific user-defined parameters for distinguishing inlier measurements from outliers. ADAPT eliminates inliers based on iterative refinement even without knowing exact noise statistics, while GNC employs a progressive approach to handle non-convex objective functions through a homotopy method.
4. Experimental Validation on Robot Perception Problems: Extensive experiments showcase the effectiveness of the proposed algorithms across several applications like mesh registration, shape alignment, and pose graph optimization. Notably, ADAPT and GNC, with their minimally tuned alternatives, demonstrate robustness against outlier rates up to 90%, consistently outperforming traditional approaches like RANSAC in both runtime efficiency and accuracy.

Theoretical and Practical Implications

The paper's theoretical contributions underscore a need to reconsider algorithmic strategies to obtain robust estimates in perceiving environments filled with erroneous data. Establishing hardness benchmarks forces a paradigm shift toward designing heuristics that effectively balance performance and computational feasibility. Practically, these insights deliver enhancements in real-time robotics systems where reliable perception is critical under adverse or unstructured sensing conditions.

Moreover, this research speaks to an overarching goal within AI and robotics: achieving autonomous perception systems that are not only resilient to outliers but adaptable to variances in noise characteristics over time—all without burdensome manual tuning. This leads to potential advancements in applications from autonomous driving to exploration missions where robustness and adaptability are paramount.

Speculation on Future Developments

Looking forward, the methodologies proposed in this paper are likely to influence the development of more robust algorithms capable of certifiable correctness across expanded sets of perception tasks. With increased reliance on AI-driven systems, the fusion of robust estimation techniques and self-certified algorithms promises future breakthroughs in artificial intelligence, particularly in autonomous systems requiring high-precision and reliable environmental understanding.

In conclusion, "Outlier-Robust Estimation: Hardness, Minimally Tuned Algorithms, and Applications" contributes significantly to the academic and practical discourse on machine perception, paving the way toward understanding and overcoming the inherent complexities associated with outlier-rich environments.