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In recent years, there has been remarkable progress in the development ofso-called certifiable perception methods, which leverage semidefinite, convexrelaxations to find global optima of perception problems in robotics. However,many of these relaxations rely on simplifying assumptions that facilitate theproblem formulation, such as an isotropic measurement noise distribution. Inthis paper, we explore the tightness of the semidefinite relaxations ofmatrix-weighted (anisotropic) state-estimation problems and reveal thelimitations lurking therein: matrix-weighted factors can cause convexrelaxations to lose tightness. In particular, we show that the semidefiniterelaxations of localization problems with matrix weights may be tight only forlow noise levels. We empirically explore the factors that contribute to thisloss of tightness and demonstrate that redundant constraints can be used toregain tightness, albeit at the expense of real-time performance. As a secondtechnical contribution of this paper, we show that the state-of-the-artrelaxation of scalar-weighted SLAM cannot be used when matrix weights areconsidered. We provide an alternate formulation and show that its SDPrelaxation is not tight (even for very low noise levels) unless specificredundant constraints are used. We demonstrate the tightness of ourformulations on both simulated and real-world data.

On Semidefinite Relaxations for Matrix-Weighted State-Estimation Problems in Robotics