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

摘要

In recent years, there has been remarkable progress in the development of so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">certifiable perception</i> methods, which leverage semidefinite, convex relaxations to find <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">global optima</i> of perception problems in robotics. However, many of these relaxations rely on simplifying assumptions that facilitate the problem formulation, such as an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">isotropic</i> measurement noise distribution. In this article, we explore the tightness of the semidefinite relaxations of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">matrix-weighted</i> (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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">redundant constraints</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">redundant constraints</i> are used. We demonstrate the tightness of our formulations on both simulated and real-world data.