  Despite enormous research interest and rapid application of federated
learning (FL) to various areas, existing studies mostly focus on supervised
federated learning under the horizontally partitioned local dataset setting.
This paper will study the unsupervised FL under the vertically partitioned
dataset setting. Accordingly, we propose the federated principal component
analysis for vertically partitioned dataset (VFedPCA) method, which reduces the
dimensionality across the joint datasets over all the clients and extracts the
principal component feature information for downstream data analysis. We
further take advantage of the nonlinear dimensionality reduction and propose
the vertical federated advanced kernel principal component analysis (VFedAKPCA)
method, which can effectively and collaboratively model the nonlinear nature
existing in many real datasets. In addition, we study two communication
topologies. The first is a server-client topology where a semi-trusted server
coordinates the federated training, while the second is the fully-decentralized
topology which further eliminates the requirement of the server by allowing
clients themselves to communicate with their neighbors. Extensive experiments
conducted on five types of real-world datasets corroborate the efficacy of
VFedPCA and VFedAKPCA under the vertically partitioned FL setting. Code is
available at: https://github.com/juyongjiang/VFedPCA-VFedAKPCA
