Space-Time Diversity in Observability and Estimation on Product Lie Groups

State estimation in coupled dynamical systems often relies on sensor quality, but little attention has been paid to the structural alignment between observation channels and the system's intrinsic dynamics. A new paper on arXiv (2605.05717) tackles this by developing a rigorous framework for analyzing spatial and temporal diversity in dynamical state estimation on product Lie groups. Drawing structural parallels to diversity gains in space-time coding, the authors—Somasundhar Venkatasubramanian, Anirudh Venkat, and Advaidh Venkat—establish three main results that extend classical observability theory beyond Euclidean state spaces.

The first result gives coupling-based necessary and sufficient conditions for cross-factor observability: a sensor local to one group factor renders another factor observable if and only if the dynamics propagate error directions across the corresponding Lie algebra components. The second result is a spatial diversity saturation theorem that identifies precisely when additional observation channels fail to expand the propagated observation subspace, revealing when extra sensors provide no structural benefit. The third result provides a time-space diversity decomposition that exactly separates instantaneous spatial information from accumulated temporal information in the estimation error covariance. Applied to planar SE(2) and spatial SE(3) navigation, the framework yields exact observability guarantees for redundant and non-redundant sensor architectures, exposing structural constraints invisible to standard rank-based analysis. This work has been submitted to IEEE for possible publication.