Global Truncated Loss Minimization for Robust and Threshold-Resilient Geometric Estimation

2026-03-15

AI Summary

This work addresses the sensitivity to outliers and strong dependence on inlier thresholds in geometric estimation by proposing the GTM framework, which for the first time systematically applies branch-and-bound (BnB) methods to globally minimize truncated losses. By integrating Lipschitz continuity bounds with the DIRECT global optimizer, the authors devise a hybrid dimensional search strategy that effectively reduces optimization dimensionality and accelerates convergence. The resulting method demonstrates significantly enhanced robustness and adaptability to inlier thresholds across a variety of geometric estimation tasks, outperforming existing approaches in both computational efficiency and estimation accuracy.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

To achieve outlier-robust geometric estimation, robust objective functions are generally employed to mitigate the influence of outliers. The widely used consensus maximization(CM) is highly robust when paired with global branch-and-bound(BnB) search. However, CM relies solely on inlier counts and is sensitive to the inlier threshold. Besides, the discrete nature of CM leads to loose bounds, necessitating extensive BnB iterations and computation cost. Truncated losses(TL), another continuous alternative, leverage residual information more effectively and could potentially overcome these issues. But to our knowledge, no prior work has systematically explored globally minimizing TL with BnB and its potential for enhanced threshold resilience or search efficiency. In this work, we propose GTM, the first unified BnB-based framework for globally-optimal TL loss minimization across diverse geometric problems. GTM involves a hybrid solving design: given an n-dimensional problem, it performs BnB search over an (n-1)-dimensional subspace while the remaining 1D variable is solved by bounding the objective function. Our hybrid design not only reduces the search space, but also enables us to derive Lipschitz-continuous bounding functions that are general, tight, and can be efficiently solved by a classic global Lipschitz solver named DIRECT, which brings further acceleration. We conduct a systematic evaluation on various BnB-based methods for CM and TL on the robust linear regression problem, showing that GTM enjoys remarkable threshold resilience and the highest efficiency compared to baseline methods. Furthermore, we apply GTM on different geometric estimation problems with diverse residual forms. Extensive experiments demonstrate that GTM achieves state-of-the-art outlier-robustness and threshold-resilience while maintaining high efficiency across these estimation tasks.

Problem

• geometric estimation
• outlier robustness
• truncated loss
• threshold resilience
• branch-and-bound

Innovation

• Truncated Loss
• Branch-and-Bound
• Threshold Resilience
• Geometric Estimation
• Global Optimization