In recent years, there has been remarkable progress in the development of so-called certifiable perception methods, which leverage semidefinite, convex relaxations to find global optima of perception problems in robotics. However, many of these relaxations rely on simplifying assumptions that facilitate the problem formulation, such as an isotropic measurement noise distribution. In this paper, we explore the tightness of the semidefinite relaxations of matrix-weighted (anisotropic) state-estimation problems and reveal the limitations lurking therein: matrix-weighted factors can cause convex relaxations to lose tightness. In particular, we show that the semidefinite relaxations of localization problems with matrix weights may be tight only for low noise levels. We empirically explore the factors that contribute to this loss of tightness and demonstrate that redundant constraints can be used to regain tightness, albeit at the expense of real-time performance. As a second technical contribution of this paper, we show that the state-of-the-art relaxation of scalar-weighted SLAM cannot be used when matrix weights are considered. We provide an alternate formulation and show that its SDP relaxation is not tight (even for very low noise levels) unless specific redundant constraints are used. We demonstrate the tightness of our formulations on both simulated and real-world data.

翻译：近年来，所谓可验证感知方法取得了显著进展，这类方法利用半定凸松弛在机器人感知问题中寻找全局最优解。然而，许多松弛方法依赖于简化假设以简化问题建模，例如各向同性测量噪声分布。本文探讨了矩阵加权（各向异性）状态估计问题的半定松弛紧致性，揭示了其中潜藏的局限性：矩阵加权因子可能导致凸松弛丧失紧致性。具体而言，我们证明具有矩阵权重的定位问题的半定松弛可能仅在低噪声水平下保持紧致。我们通过实验探究了导致紧致性丧失的因素，并证明冗余约束可恢复紧致性，但会牺牲实时性能。作为本文的第二项技术贡献，我们证明当考虑矩阵权重时，标量加权SLAM的现有最优松弛无法适用。我们提出了一种替代公式，并证明其SDP松弛（即使在极低噪声水平下）也不具有紧致性，除非使用特定的冗余约束。我们通过在仿真数据和真实数据上的实验验证了所提公式的紧致性。