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. To better understand this issue, we introduce a theoretical connection between the posterior uncertainty of the state estimate and the dual variable of the convex relaxation. With this connection in mind, we empirically explore the factors that contribute to this loss of tightness and demonstrate that redundant constraints can be used to regain it. 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，所谓的可验证感知方法取得了显著进展，这些方法利用半定凸松弛来寻找机器人感知问题的全局最优解。然而，许多此类松弛依赖于简化假设（例如各向同性测量噪声分布）以简化问题表述。本文探讨了矩阵加权（各向异性）状态估计问题的半定松弛的紧致性，并揭示了其中隐藏的局限性：矩阵加权因子可能导致凸松弛失去紧致性。具体而言，我们表明，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才是紧致的。为更好地理解这一问题，我们引入了状态估计后验不确定性与其凸松弛对偶变量之间的理论联系。基于此联系，我们实证探究了导致这种紧致性丧失的因素，并证明冗余约束可用于重新恢复紧致性。作为本文的第二个技术贡献，我们表明，当考虑矩阵权重时，标量加权SLAM的现有最先进松弛无法使用。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）也不是紧致的，除非使用特定的冗余约束。我们在模拟和真实数据上展示了我们公式的紧致性。