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