Reduced-Dimension Design of MIMO Over-the-Air Computing for Data Aggregation in Clustered IoT Networks

Abstract

One basic operation of Internet-of-Things (IoT) networks is aggregating distributed sensing data collected over wireless channels to compute a desired function, called wireless data aggregation (WDA).In the presence of dense sensors, low-latency WDA poses a design challenge for high-mobility or mission critical IoT applications. A technology called over-the-air computing (AirComp) can dramatically reduce the WDA latency by aggregating distributed data "over-the-air" using the waveformsuperposition 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 highdimensional but low-rank MIMO channels makes it important to reduce channel/signal dimensionality in AirComp to avoid exposure to noise from channel null-spaces. The design challenge lies in the integration of simultaneous dimension-reduction and joint-equalization (without decoupling) of many MIMO channels with correlation and heterogeneous ranks. By tackling the challenge, we develop in this work a framework of reduced-dimension MIMO AirComp. The key component is decomposed aggregation beamforming (DAB) for the AP. Consider the case of separable channel clusters with non-overlapping angle-of-arrival (AoA) ranges. The optimal DAB is proved to have the architecture 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. As part of the said framework, we also design efficient algorithms for rank optimization of individual DAB components and channel feedback leveraging the AirComp principle. The proposed framework is shown by simulation to substantially reduce AirComp error compared with the existing design without considering channel structures.

Broadband Analog Aggregation for Low-Latency Federated Edge Learning

“…Advancing beyond scalar-valued function computation, the latest trend in the area also explores multipleinput-mutiple-output (MIMO) techniques to enable vector-valued function computation [25]- [27], referred as MIMO AirComp. In particular, a comprehensive framework for MIMO AirComp that consists of beamforming optimization and a matching limited-feedback design is proposed in [25].…”

“…In particular, a comprehensive framework for MIMO AirComp that consists of beamforming optimization and a matching limited-feedback design is proposed in [25]. The framework was extended in subsequent work to wirelessly-powered AirComp system [26], where the beamformer was jointly optimized with the wireless power control to further reduce the AirComp distortion, and massive MIMO AirComp system [27], where a reduced-dimension two-tier beamformer design was developed by exploiting the clustered channel structure to reduce the channel-feedback overhead and signal processing complexity. It is also worth mentioning that, while AirComp is mostly deployed in computation-centric sensor networks as discussed above, the AirComp operation has been also leveraged in rate-maximization cellular systems such as two-way relaying [28] and MIMO lattice decoding [29].…”

Optimal Power Control for Over-the-Air Computation

“…To solve problem (14), we need to remove the "min" operation to simplify the derivation. To this end, we find it convenient to adopt a divide-and-conquer approach that divides the feasible set of problem (14), namely {η ≥ 0}, into K + 1 intervals. Note that each of them is defined as…”

“…It is evident that the optimal solution to problem (45) is η * 0 =P 1 |h 1 | 2 , therefore, η * 0 =P 1 |h 1 | 2 can achieve a lower MSE value than any η <P 1 |h 1 | 2 . As a result, it must hold that η * ≥P 1 |h 1 | 2 for problem (14). Accordingly, it follows from (13) that device 1 should always transmit with full power, i.e., p * 1 =P k .…”

“…A so-called "AirShare" design in [12] was presented for resolving the synchronization problem in distributed transmission, in which the FC can broadcast a shared clock to all devices. More recently, the multiple-input-multiple-output (MIMO) technique was exploited in [13], [14] to enable vector-valued AirComp targeting multi-modal sensing, in which zero-forcing precoding at sensors and aggregation beamforming at FC are jointly designed for minimizing the computation error measured by the mean square error (MSE).…”