In paper 'Local Identification of Overcomplete Dictionaries', the authors mentioned that another paper 1) shows the K-SVD minimisation principle can locally identify overcomplete tight dictionaries; 2) shows that for S = 1 the K-SVD criterion (3) can only identify
the underlying dictionary from sparse random mixtures to arbitrary precision (given enough
training samples) if this dictionary is tight and it is conjectured that the same holds for
S ≥ 1. You've also read that paper. Provide the full name of that paper.