  This article treats the problem of learning a dictionary providing sparse
representations for a given signal class, via $\ell_1$-minimisation. The
problem can also be seen as factorising a $\ddim \times \nsig$ matrix $Y=(y_1
>... y_\nsig), y_n\in \R^\ddim$ of training signals into a $\ddim \times
\natoms$ dictionary matrix $\dico$ and a $\natoms \times \nsig$ coefficient
matrix $\X=(x_1... x_\nsig), x_n \in \R^\natoms$, which is sparse. The exact
question studied here is when a dictionary coefficient pair $(\dico,\X)$ can be
recovered as local minimum of a (nonconvex) $\ell_1$-criterion with input
$Y=\dico \X$. First, for general dictionaries and coefficient matrices,
algebraic conditions ensuring local identifiability are derived, which are then
specialised to the case when the dictionary is a basis. Finally, assuming a
random Bernoulli-Gaussian sparse model on the coefficient matrix, it is shown
that sufficiently incoherent bases are locally identifiable with high
probability. The perhaps surprising result is that the typically sufficient
number of training samples $\nsig$ grows up to a logarithmic factor only
linearly with the signal dimension, i.e. $\nsig \approx C \natoms \log
\natoms$, in contrast to previous approaches requiring combinatorially many
samples.
