  This work considers two related learning problems in a federated attack prone
setting: federated principal components analysis (PCA) and federated low rank
column-wise sensing (LRCS). The node attacks are assumed to be Byzantine which
means that the attackers are omniscient and can collude. We introduce a novel
provably Byzantine-resilient communication-efficient and sampleefficient
algorithm, called Subspace-Median, that solves the PCA problem and is a key
part of the solution for the LRCS problem. We also study the most natural
Byzantine-resilient solution for federated PCA, a geometric median based
modification of the federated power method, and explain why it is not useful.
Our second main contribution is a complete alternating gradient descent (GD)
and minimization (altGDmin) algorithm for Byzantine-resilient horizontally
federated LRCS and sample and communication complexity guarantees for it.
Extensive simulation experiments are used to corroborate our theoretical
guarantees. The ideas that we develop for LRCS are easily extendable to other
LR recovery problems as well.
