  Online Matrix Factorization (OMF) is a fundamental tool for dictionary
learning problems, giving an approximate representation of complex data sets in
terms of a reduced number of extracted features. Convergence guarantees for
most of the OMF algorithms in the literature assume independence between data
matrices, and the case of dependent data streams remains largely unexplored. In
this paper, we show that a non-convex generalization of the well-known OMF
algorithm for i.i.d. stream of data in \citep{mairal2010online} converges
almost surely to the set of critical points of the expected loss function, even
when the data matrices are functions of some underlying Markov chain satisfying
a mild mixing condition. This allows one to extract features more efficiently
from dependent data streams, as there is no need to subsample the data sequence
to approximately satisfy the independence assumption. As the main application,
by combining online non-negative matrix factorization and a recent MCMC
algorithm for sampling motifs from networks, we propose a novel framework of
Network Dictionary Learning, which extracts ``network dictionary patches' from
a given network in an online manner that encodes main features of the network.
We demonstrate this technique and its application to network denoising problems
on real-world network data.
