🤖 AI Summary
In robot state estimation, anisotropic (matrix-weighted) noise causes conventional convex relaxations to lose tightness, thereby invalidating certification guarantees. Method: This paper identifies the fundamental mechanism by which matrix weighting undermines the tightness of semidefinite relaxation (SDR), establishes a theoretical link between posterior uncertainty and the certificate matrix, and proposes a novel paradigm—incorporating geometric redundancy constraints—to restore tightness. It further constructs the first tightable semidefinite programming (SDP) formulation for matrix-weighted SLAM. Contribution/Results: We theoretically prove that the proposed SDP admits a tightness bound under low-noise conditions. Extensive simulations and real-world experiments demonstrate that redundancy constraints significantly improve tightness rates; moreover, the new formulation maintains tightness even under extremely low noise, outperforming scalar-weighted approaches in both certification reliability and estimation accuracy.
📝 Abstract
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 article, 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 certificate matrix obtained via 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 article, we show that the state-of-the-art relaxation of scalar-weighted simultaneous localization and mapping cannot be used when matrix weights are considered. We provide an alternate formulation and show that its semidefinite program 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.