  One basic operation of Internet-of-Things (IoT) networks is aggregating
distributed sensing-data over wireless-channels for function-computation,
called wireless-data-aggregation (WDA). Targeting dense sensors, a recently
developed technology called over-the-air computing (AirComp) can dramatically
reduce the WDA latency by aggregating distributed data "over-the-air" using the
waveform-superposition property of a multi-access channel. In this work, we
design multiple-input-multiple-output (MIMO) AirComp for computing a
vector-valued function in a clustered IoT network with multi-antenna sensors
forming clusters and a multi-antenna access-point (AP) performing WDA. The
resultant high-dimensional but low-rank MIMO channels makes it important to
reduce channel or signal dimensionality in AirComp to avoid exposure to noise
from channel null-spaces. Motivated by this, we develop a framework of
reduced-dimension MIMO AirComp, featuring decomposed-aggregation-beamforming
(DAB). Consider the case of separable channel-clusters with non-overlapping
angle-of-arrival ranges. The optimal DAB has the structure where
inner-components extract the dominant eigen-spaces of corresponding
channel-clusters and outer-components jointly equalize the resultant
low-dimensional channels. Consider the more complex case of inseparable
clusters. We propose a suboptimal DAB design where the inner-component performs
both dimension-reduction and joint-equalization over clustered-channel
covariance matrices and the outer-component jointly equalizes the small-scale
fading-channels. Furthermore, efficient algorithms for rank-optimization of
individual DAB components and channel-feedback leveraging the AirComp principle
are developed.
