Estimation Contracts for Outlier-Robust Geometric Perception

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 paper 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 regressions (applicable to the low-outlier case with << 50% outliers) and list-decodable regression (applicable to the high-outlier case 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 new algorithms, we provide conditions on the input measurements under which modern estimation algorithms 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 paper 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.