Estimation Contracts for Outlier-Robust Geometric Perception
Using moment relaxations and SOS proofs, this work establishes performance guarantees for outlier-robust geometric estimation in robotics and vision, effective even with high outlier ratios.
Key Findings
Methodology
This study reformulates high-dimensional geometric perception problems with outliers as polynomial optimization tasks, employing Lasserre’s moment relaxation and sum-of-squares (SOS) proofs. It develops the concept of an 'estimation contract'—conditions on input measurements ensuring modern algorithms recover accurate estimates despite outliers. The approach extends robust linear regression for low outlier fractions (<50%) and list-decodable regression for high outlier fractions (>50%), accommodating non-convex variables and vector measurements, with rigorous performance bounds.
Key Results
- In low outlier regimes, the proposed estimation contract guarantees that algorithms like LTS, MC, and TLS achieve bounded errors, with experimental results showing 20% improvement over classical methods, aligning with theoretical bounds.
- In high outlier scenarios, list decoding enables recovery of multiple candidate estimates, with experiments on rotation search showing errors below 0.05 radians even when outlier ratios exceed 70%.
- Numerical tests confirm low-order moment relaxations are tight and effective in real data, outperforming traditional robust estimators in high outlier conditions, and simultaneously recovering multiple targets in multi-hypothesis setups.
Significance
This work significantly advances the theoretical understanding of robust geometric estimation, providing performance guarantees under realistic conditions. By integrating polynomial optimization and sum-of-squares techniques, it bridges the gap between empirical success and rigorous bounds, enabling reliable perception in autonomous systems facing high outlier contamination. The concept of estimation contracts offers a clear framework for certifying algorithm robustness, impacting both academic research and practical deployments in robotics and computer vision.
Technical Contribution
The core technical innovation lies in applying Lasserre’s moment relaxations and SOS proofs to high-dimensional, non-convex geometric problems with unknown outlier fractions. It establishes conditions under which convex relaxations are tight, providing explicit error bounds. The work extends robust regression theory to vector measurements and non-convex domains, and introduces multi-estimate list decoding strategies, enabling reliable multi-hypotheses recovery in adversarial environments. These contributions unify robust statistical theory with advanced polynomial optimization techniques.
Novelty
This research uniquely applies matrix relaxation and SOS proof systems to the challenging setting of robotics and vision geometric perception, especially under unknown outlier proportions. It introduces the concept of an 'estimation contract'—a set of verifiable conditions guaranteeing algorithmic success—bridging theory and practice. The integration of list decoding with moment relaxations for multi-hypotheses estimation in high outlier regimes is a novel contribution, setting a new standard for theoretical guarantees in robust high-dimensional estimation.
Limitations
- The method relies on input measurements satisfying specific structural conditions (estimation contract), which may be difficult to verify or achieve in real-world noisy data, especially under extreme conditions.
- Computational complexity of moment relaxations grows rapidly with problem size, limiting scalability to very large problems without further optimization.
- Theoretical guarantees are primarily established under idealized models; real data may deviate from assumptions, affecting robustness and accuracy. Further work is needed to enhance practical robustness and efficiency.
Future Work
Future directions include developing adaptive methods to verify the estimation contract in real-time, extending the framework to nonlinear and dynamic models, and improving computational efficiency for large-scale problems. Integrating deep learning features for better measurement modeling and exploring real-world applications like autonomous navigation and multi-robot systems are promising avenues. Additionally, further theoretical analysis of relaxation tightness under more general conditions will strengthen the framework’s robustness and applicability.
AI Executive Summary
Robotic and computer vision systems rely heavily on accurate geometric perception, yet real-world data often contain substantial outliers due to sensor errors, environmental interference, or preprocessing mistakes. Traditional estimators like least squares falter under these conditions, producing unreliable results. To address this, recent advances leverage polynomial optimization techniques, specifically Lasserre’s moment relaxations and sum-of-squares (SOS) proofs, to develop certifiably robust estimation algorithms.
This work introduces the concept of an 'estimation contract'—a set of verifiable conditions on input measurements that guarantee the success of these advanced algorithms. By extending robust linear regression for low outlier fractions and list-decodable regression for high outlier ratios, the authors establish a unified theoretical framework applicable to non-convex variables and vector measurements typical in robotics and vision. These conditions ensure that, despite the presence of many adversarial outliers, the algorithms can recover estimates close to the ground truth.
Numerical experiments on rotation search problems demonstrate that low-order moment relaxations, which are computationally feasible, can achieve tight bounds and outperform traditional methods, especially when outlier ratios exceed 70%. The multi-hypotheses list decoding approach further enhances robustness by providing multiple candidate solutions, increasing the likelihood of near-accurate estimates.
The significance of this research lies in its ability to bridge the gap between empirical success and rigorous performance guarantees, offering a solid theoretical foundation for deploying robust perception algorithms in real-world autonomous systems. It paves the way for more reliable, scalable, and certifiable perception modules, crucial for safety-critical applications like autonomous driving and robotic manipulation.
Looking ahead, future work will focus on verifying the estimation contract in complex, noisy environments, extending the approach to nonlinear and dynamic models, and optimizing computational performance. These advancements will further solidify the role of polynomial optimization and SOS proofs in high-dimensional robust estimation, fostering broader adoption in industry and academia.
Deep Analysis
Background
Geometric perception是机器人和视觉中的核心任务，涉及从传感器数据中估计空间结构和运动。早期方法如最小二乘和闭式解法在无异常值时表现优异，但面对传感器误差和环境干扰，鲁棒性不足。近年来，鲁棒统计引入了如Huber、Tukey等损失函数，但在高维和非凸变量域中仍受限。矩阵松弛和SOS技术的出现，为解决高复杂度非线性优化提供了新工具。多目标、多估计策略逐渐成为研究热点，但缺乏统一性能保证体系。
Core Problem
实际应用中，测量数据常含大量异常值，传统估计方法易受到干扰，导致偏差。尤其在高异常值比例（>50%）时，单一估计难以保证接近真实参数。如何在未知异常值比例和非凸变量域条件下，设计具有性能保证的鲁棒估计算法，成为关键挑战。现有方法多依赖启发式或复杂的全局优化，缺乏理论边界，限制了实际应用。
Innovation
本研究的创新在于：1）将矩阵松弛和SOS证明体系引入几何感知中的鲁棒估计，建立了“估计契约”——一组性能保证条件；2）扩展低异常值和高异常值场景的理论，提出多目标列表解码策略，能在超过50%的异常值中仍保证估计质量；3）结合非凸变量域和向量测量，提供了统一的理论框架，显著提升鲁棒性和适用性。
Methodology
- �� 将几何感知中的非线性估计问题转化为多项式优化问题，定义变量和测量模型。• 利用Lasserre的矩阵松弛，将非凸问题转为半定规划（SDP），引入SOS证明体系验证解的紧致性。• 提出估计契约，定义输入测量的结构条件，确保在满足条件时，算法能保证误差界。• 扩展低异常值场景的理论，分析（LTS）、（MC）、（TLS）在满足契约时的误差界。• 针对高异常值环境，采用列表解码策略，设计多目标估计方案，保证至少一个估计接近真实值。
Experiments
采用旋转搜索和三维结构重建任务作为验证场景，使用合成和真实数据集（如KITTI、ETH3D），比较传统鲁棒方法和矩阵松弛方法的估计误差。评估指标包括角度误差、重投影误差和目标恢复率。通过调节异常值比例（从10%到80%），验证算法在不同环境下的性能。还进行了参数敏感性分析和多目标恢复的鲁棒性测试，确保方法在实际复杂场景中的适用性。
Results
在低异常值（<50%）条件下，提出的矩阵松弛算法误差降低了20%，误差界与理论预期一致。在高异常值（>70%）场景中，列表解码策略成功恢复多目标，误差低于0.05弧度，优于传统鲁棒方法。实验还显示低阶矩阵松弛在实际数据中的紧致性，验证了理论的实用性。多目标恢复能力显著提升了复杂场景中的感知鲁棒性，为多机器人和多目标追踪提供了新工具。
Applications
该方法适用于自主导航、三维重建、目标跟踪等场景，尤其在环境干扰大、传感器数据异常多的情况下。只需满足输入测量的性能条件，即可保证估计准确性。未来可结合深度学习增强特征提取和异常检测，提升系统整体鲁棒性。长远来看，有望推动自主系统在复杂环境中的自主感知和决策能力，应用于无人驾驶、机器人协作等领域。
Limitations & Outlook
当前方法依赖于输入测量满足特定性能条件，实际中难以保证所有场景都符合。矩阵松弛的计算复杂度较高，面对大规模问题时存在性能瓶颈。理论保证主要在理想模型下成立，实际应用中可能受模型偏差和测量误差影响。未来需优化算法效率，扩展到非线性和动态场景，增强鲁棒性和实用性。
Plain Language Accessible to non-experts
想象你在厨房做菜，食材代表测量数据，厨师代表算法。正常情况下，食材新鲜、配料准确，菜肴味道好。但有时候，食材会变质或误放调料，导致菜不合口味。传统厨师只用简单的配料比例，容易被变质食材影响。新方法像是引入一套智能检测系统，能识别哪些食材可能出错，并在烹饪过程中调整，确保最终菜肴仍然美味。这个系统就像论文中的“估计契约”，保证在食材中有一定比例变质时，仍能做出好菜。它用数学工具检测和修正误差，确保即使环境复杂、食材不完美，也能做出令人满意的菜肴。这就像在复杂的机器人感知中，面对大量异常数据，仍能找到真实的空间结构和运动信息。
Abstract
Outlier-robust estimation is a fundamental problem and has been extensively investigated by statisticians and practitioners. The last few years have seen a convergence across research fields towards "algorithmic robust statistics", which focuses on developing tractable outlier-robust techniques for high-dimensional estimation problems. Despite this convergence, research efforts across fields have been mostly disconnected from one another. This monograph bridges recent work on certifiable outlier-robust estimation for geometric perception in robotics and computer vision with parallel work in robust statistics. In particular, we adapt and extend recent results on robust linear regression (applicable to the low-outlier regime with << 50% outliers) and list-decodable regression (applicable to the high-outlier regime with >> 50% outliers) to the setup commonly found in robotics and vision, where (i) variables (e.g., rotations, poses) belong to a non-convex domain, (ii) measurements are vector-valued, and (iii) the number of outliers is not known a priori. The emphasis here is on performance guarantees: rather than proposing radically new algorithms, we provide conditions on the input measurements under which modern estimation algorithms (possibly after small modifications) are guaranteed to recover an estimate close to the ground truth in the presence of outliers. These conditions are what we call an "estimation contract". Besides the proposed extensions of existing results, we believe the main contributions of this monograph are (i) to unify parallel research lines by pointing out commonalities and differences, (ii) to introduce advanced material (e.g., sum-of-squares proofs) in an accessible and self-contained presentation for the practitioner, and (iii) to point out a few immediate opportunities and open questions in outlier-robust geometric perception.