Dictionary identifiability from few training samples

This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via ℓ¹ minimisation. The problem is to identify a dictionary Φ from a set of training samples Y knowing that Y = ΦX for some coefficient matrix X. Using a characterisation of coefficient matrices X that allow to recover any orthonormal basis (ONB) as a local minimum of an ℓ¹ minimisation problem, it is shown that certain types of sparse random coefficient matrices will ensure local identifiability of the ONB with high probability, for a number of training samples which essentially grows linearly with the signal dimension.