  We aim to solve the problem of spatially localizing composite instructions
referring to space: space grounding. Compared to current instance grounding,
space grounding is challenging due to the ill-posedness of identifying
locations referred to by discrete expressions and the compositional ambiguity
of referring expressions. Therefore, we propose a novel probabilistic
space-grounding methodology (LINGO-Space) that accurately identifies a
probabilistic distribution of space being referred to and incrementally updates
it, given subsequent referring expressions leveraging configurable polar
distributions. Our evaluations show that the estimation using polar
distributions enables a robot to ground locations successfully through $20$
table-top manipulation benchmark tests. We also show that updating the
distribution helps the grounding method accurately narrow the referring space.
We finally demonstrate the robustness of the space grounding with simulated
manipulation and real quadruped robot navigation tasks. Code and videos are
available at https://lingo-space.github.io.
