  In recent work, we proposed a distributed Banach-Picard iteration (DBPI) 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 DBPI 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 DBPI, we derive two distributed algorithms
- distributed EM and distributed PCA - whose LLC guarantees follow from those
that we proved for the DBPI. 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.
