Distributed Picard Iteration: Application to Distributed EM and Distributed PCA

In recent work, we proposed a distributed Picard iteration (DPI) that allows a set of agents, linked by a communication network, to find a fixed point of a locally contractive (LC) map that is the average of individual maps held by said agents. In this work, we build upon the DPI and its local linear convergence (LLC) guarantees to make several contributions. We show that Sanger's algorithm for principal component analysis (PCA) corresponds to the iteration of an LC map that can be written as the average of local maps, each map known to each agent holding a subset of the data. Similarly, we show that a variant of the expectation-maximization (EM) algorithm for parameter estimation from noisy and faulty measurements in a sensor network can be written as the iteration of an LC map that is the average of local maps, each available at just one node. Consequently, via the DPI, we derive two distributed algorithms - distributed EM and distributed PCA - whose LLC guarantees follow from those that we proved for the DPI. The verification of the LC condition for EM is challenging, as the underlying operator depends on random samples, thus the LC condition is of probabilistic nature.