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This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via $ell^1$ minimisation, or more precisely the problem of identifying a dictionary $dico$ from a set of training samples $Y$ knowing that $Y = dico X$ for some coefficient matrix $X$. It provides a characterisation of coefficient matrices $X$ that allow to recover any orthonormal basis (ONB) as a local minimum of an $ell^1$ minimisation problem. Based on this characterisation it is shown that certain types of sparse random coefficient matrices will ensure local identifiability of the ONB with high probability.