[Paper Review] Outlier-Robust Estimation: Hardness, Minimally Tuned Algorithms, and Applications

Pasquale Antonante, Vasileios Tzoumas+2 more|arXiv (Cornell University)|Jul 29, 2020

Anomaly Detection Techniques and Applications4 citations

TL;DR

This paper introduces two unified formulations for outlier-robust estimation—Generalized Maximum Consensus (G-MC) and Generalized Truncated Least Squares (G-TLS)—and proves that outlier-robust estimation is inapproximable in the worst case. It proposes minimally tuned algorithms, ADAPT and GNC, which dynamically separate inliers from outliers without requiring prior noise statistics, achieving real-time performance and robustness up to 80–90% outliers in robotics applications like mesh registration, shape alignment, and pose graph optimization.

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.

Motivation & Objective

• Address the fundamental challenge of outlier-robust estimation in robotics and vision, where data associations are frequently corrupted by outliers due to sensor noise or incorrect matching.
• Investigate the computational hardness of outlier-robust estimation, proving it is inapproximable even with quasi-polynomial time algorithms.
• Develop minimally tuned algorithms that do not require prior knowledge of inlier noise statistics or user-specified thresholds, enabling robust performance across diverse robotics applications.
• Evaluate the proposed algorithms on real-world perception tasks, demonstrating superior performance and robustness compared to state-of-the-art methods like RANSAC and DCS.

Proposed method

• Introduce two unifying formulations: Generalized Maximum Consensus (G-MC) for sparse outlier rejection and Generalized Truncated Least Squares (G-TLS) for continuous outlier handling.
• Propose ADAPT, a combinatorial algorithm based on adaptive trimming, for solving G-MC problems by iteratively identifying and removing the most inconsistent measurements.
• Propose GNC (Graduated Non-Convexity), a homotopy-based method for G-TLS that gradually enforces robustness by modifying the cost function's shape through a sequence of non-convex subproblems.
• Design minimally tuned variants, ADAPT-MinT and GNC-MinT, that dynamically estimate inlier thresholds using statistical tests such as the Cramér–von Mises test on residual distributions.
• Implement a clustering separation subroutine in ADAPT-MinT that uses the diameter of sorted residuals to detect natural splits between inlier and outlier clusters.
• Use a chi-squared goodness-of-fit test (Chi2Fit) in GNC-MinT to assess whether the squared residuals match the expected chi-squared distribution under the assumed inlier model, enabling automatic threshold selection.

Experimental results

Research questions

• RQ1 What are the fundamental computational limits of outlier-robust estimation, and is it possible to approximate the optimal set of inliers in polynomial or quasi-polynomial time?
• RQ2 Can we design general-purpose algorithms for outlier-robust estimation that do not require prior knowledge of inlier noise statistics or user-specified thresholds?
• RQ3 How do minimally tuned variants of ADAPT and GNC perform compared to state-of-the-art methods like RANSAC and DCS in real-world robotics applications?
• RQ4 To what extent can ADAPT and GNC handle high outlier rates (e.g., 80–90%) while maintaining accuracy and real-time performance?
• RQ5 How do the proposed algorithms balance true positive and false positive outlier detection rates across varying outlier levels in different perception tasks?

Key findings

• The paper proves that outlier-robust estimation is inapproximable in the worst case, even with quasi-polynomial time algorithms, establishing a fundamental hardness limit.
• ADAPT and GNC achieve real-time performance and outperform RANSAC in all evaluated tasks, maintaining robustness up to 80–90% outlier rates in mesh registration, shape alignment, and pose graph optimization.
• The minimally tuned variants, ADAPT-MinT and GNC-MinT, achieve competitive performance without requiring user-defined noise bounds or kernel sizes, outperforming tuned baselines in some settings.
• GNC and GNC-MinT achieve the highest true positive rates (over 90%) and lowest false positive rates (10–20%) across most datasets, with only minor degradation in high-outlier regimes.
• In the CSAIL and Garage 3D SLAM datasets, GNC-MinT maintains high accuracy even at 90% outlier rates, demonstrating strong resilience and minimal tuning.
• The Chi2Fit test in GNC-MinT successfully identifies inlier distributions by comparing residual squared magnitudes to a chi-squared distribution, enabling automatic thresholding without prior knowledge.