  In this work, we present a new approach for the distributed computation of
the PARAFAC decomposition of a third-order tensor across a network of
collaborating nodes. We are interested in the case where the overall data
gathered across the network can be modeled as a data tensor admitting an
essentially unique PARAFAC decomposition, while each node only observes a
sub-tensor with not necessarily enough diversity so that identifiability
conditions are not locally fulfilled at each node. In this situation,
conventional (centralized) tensor based methods cannot be applied individually
at each node. By allowing collaboration between neighboring nodes of the
network, we propose distributed versions of the alternating least squares (ALS)
and Levenberg-Marquardt (LM) algorithms for the in-network estimation of the
factor matrices of a third-order tensor. We assume that one of the factor
matrices contains parameters that are local to each node, while the two
remaining factor matrices contain global parameters that are common to the
whole network. The proposed algorithms combine the estimation of the local
factors with an in-network computation of the global factors of the PARAFAC
decomposition using average consensus over graphs. They emulate their
centralized counterparts in the case of ideal data exchange and ideal consensus
computations. The performance of the proposed algorithms are evaluated in both
ideal and imperfect cases.
