  Recently years, the attempts on distilling mobile data into useful knowledge
has been led to the deployment of machine learning algorithms at the network
edge. Principal component analysis (PCA) is a classic technique for extracting
the linear structure of a dataset, which is useful for feature extraction and
data compression. In this work, we propose the deployment of distributed PCA
over a multi-access channel based on the algorithm of stochastic gradient
descent to learn the dominant feature space of a distributed dataset at
multiple devices. Over-the-air aggregation is adopted to reduce the
multi-access latency, giving the name over-the-air PCA. The novelty of this
design lies in exploiting channel noise to accelerate the descent in the region
around each saddle point encountered by gradient descent, thereby increasing
the convergence speed of over-the-air PCA. The idea is materialized by
proposing a power-control scheme which detects the type of descent region and
controlling the level of channel noise accordingly. The scheme is proved to
achieve a faster convergence rate than in the case without power control.
