  In this work, we study the problem of semantic communication and inference,
in which a student agent (i.e. mobile device) queries a teacher agent (i.e.
cloud sever) to generate higher-order data semantics living in a simplicial
complex. Specifically, the teacher first maps its data into a k-order
simplicial complex and learns its high-order correlations. For effective
communication and inference, the teacher seeks minimally sufficient and
invariant semantic structures prior to conveying information. These minimal
simplicial structures are found via judiciously removing simplices selected by
the Hodge Laplacians without compromising the inference query accuracy.
Subsequently, the student locally runs its own set of queries based on a masked
simplicial convolutional autoencoder (SCAE) leveraging both local and remote
teacher's knowledge. Numerical results corroborate the effectiveness of the
proposed approach in terms of improving inference query accuracy under
different channel conditions and simplicial structures. Experiments on a
coauthorship dataset show that removing simplices by ranking the Laplacian
values yields a 85% reduction in payload size without sacrificing accuracy.
Joint semantic communication and inference by masked SCAE improves query
accuracy by 25% compared to local student based query and 15% compared to
remote teacher based query. Finally, incorporating channel semantics is shown
to effectively improve inference accuracy, notably at low SNR values.
