Gribonval, RemiSchnass, Karin2008-02-252008-02-252008-02-252008https://infoscience.epfl.ch/handle/20.500.14299/19008This 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.lts2sparse representationdictionary learningbasis learningrecovery conditionrandom coefficientstext::conference output::conference proceedings::conference paper