  A popular approach within the signal processing and machine learning
communities consists in modelling signals as sparse linear combinations of
atoms selected from a learned dictionary. While this paradigm has led to
numerous empirical successes in various fields ranging from image to audio
processing, there have only been a few theoretical arguments supporting these
evidences. In particular, sparse coding, or sparse dictionary learning, relies
on a non-convex procedure whose local minima have not been fully analyzed yet.
In this paper, we consider a probabilistic model of sparse signals, and show
that, with high probability, sparse coding admits a local minimum around the
reference dictionary generating the signals. Our study takes into account the
case of over-complete dictionaries and noisy signals, thus extending previous
work limited to noiseless settings and/or under-complete dictionaries. The
analysis we conduct is non-asymptotic and makes it possible to understand how
the key quantities of the problem, such as the coherence or the level of noise,
can scale with respect to the dimension of the signals, the number of atoms,
the sparsity and the number of observations.
