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.

翻译：为实现异常值鲁棒的几何估计，通常采用鲁棒目标函数以削弱异常值的影响。广泛使用的一致性最大化(CM)方法在与全局分支定界(BnB)搜索结合时表现出极强的鲁棒性。然而，CM仅依赖于内点计数且对内点阈值敏感。此外，CM的离散特性导致其边界松弛，需要大量BnB迭代和计算成本。截断损失(TL)作为另一种连续替代方案，能更有效地利用残差信息，并有望克服上述问题。但据我们所知，尚无研究系统性地探索基于Bnb的TL全局最小化及其在提升阈值弹性或搜索效率方面的潜力。本研究提出GTM——首个基于Bnb的统一框架，用于跨多种几何问题的全局最优TL损失最小化。GTM采用混合求解设计：对于n维问题，其在(n-1)维子空间执行Bnb搜索，而剩余的一维变量通过目标函数定界求解。该混合设计不仅缩减了搜索空间，还使我们能够推导出通用、紧致且满足Lipschitz连续性的定界函数，这些函数可通过经典全局Lipschitz求解器DIRECT高效求解，从而带来进一步加速。我们在鲁棒线性回归问题上对基于Bnb的CM与TL方法进行了系统评估，结果表明相较于基线方法，GTM具有显著的阈值弹性和最高效率。此外，我们将GTM应用于具有不同残差形式的多种几何估计问题。大量实验证明，GTM在这些估计任务中实现了最先进的异常值鲁棒性与阈值弹性，同时保持了高效率。