Revisiting maximal response-based local identification of overcomplete dictionaries

This paper revisits the problem of recovery of an overcomplete dictionary in a local neighborhood from training samples using the so-called maximal response criterion (MRC). While it is known in the literature that MRC can be used for asymptotic exact recovery of a dictionary in a local neighborhood, those results do not allow for linear (in the ambient dimension) scaling of sparsity levels in signal representations. In this paper, a new proof technique is leveraged to establish that MRC can in fact handle linear sparsity (modulo a logarithmic factor) of signal representations. While the focus of this work is on asymptotic exact recovery, the same ideas can be used in a straightforward manner to strengthen the original MRC-based results involving noisy observations and finite number of training samples.