  General-purpose mobile robots need to complete tasks without exact human
instructions. Large language models (LLMs) is a promising direction for
realizing commonsense world knowledge and reasoning-based planning.
Vision-language models (VLMs) transform environment percepts into
vision-language semantics interpretable by LLMs. However, completing complex
tasks often requires reasoning about information beyond what is currently
perceived. We propose latent compositional semantic embeddings z* as a
principled learning-based knowledge representation for queryable
spatio-semantic memories. We mathematically prove that z* can always be found,
and the optimal z* is the centroid for any set Z. We derive a probabilistic
bound for estimating separability of related and unrelated semantics. We prove
that z* is discoverable by iterative optimization by gradient descent from
visual appearance and singular descriptions. We experimentally verify our
findings on four embedding spaces incl. CLIP and SBERT. Our results show that
z* can represent up to 10 semantics encoded by SBERT, and up to 100 semantics
for ideal uniformly distributed high-dimensional embeddings. We demonstrate
that a simple dense VLM trained on the COCO-Stuff dataset can learn z* for 181
overlapping semantics by 42.23 mIoU, while improving conventional
non-overlapping open-vocabulary segmentation performance by +3.48 mIoU compared
with a popular SOTA model.
