FT-BSP: Focused Topological Belief Space Planning

At its core, decision making under uncertainty can be regarded as sorting candidate actions according to a certain objective. While finding the optimal solution directly is computationally expensive, other approaches that produce the same ordering of candidate actions, will result in the same selection. With this motivation in mind, we present a computationally efficient approach for the focused belief space planning (BSP) problem, where reducing the uncertainty of only a predefined subset of variables is of interest. Our approach uses topological signatures, defined over the topology induced from factor graph representations of posterior beliefs, to rank candidate actions. In particular, we present two such signatures in the context of information theoretic focused decision making problems. We derive error bounds, with respect to the optimal solution, and prove that one of these signatures converges to the true optimal solution. We also derive a second set of bounds, which is more conservative, but is only a function of topological aspects and can be used online. We introduce the Von Neumann graph entropy for the focused case, which is based on weighted node degrees, and show that it supports incremental update. We then analyze our approach under two different settings, measurement selection and active focused 2D pose SLAM.