  In this paper, it is proved that dictionary learning and sparse
representation is invariant to a linear transformation. It subsumes the special
case of transforming/projecting the data into a discriminative space. This is
important because recently, supervised dictionary learning algorithms have been
proposed, which suggest to include the category information into the learning
of dictionary to improve its discriminative power. Among them, there are some
approaches that propose to learn the dictionary in a discriminative projected
space. To this end, two approaches have been proposed: first, assigning the
discriminative basis as the dictionary and second, perform dictionary learning
in the projected space. Based on the invariance of dictionary learning to any
transformation in general, and to a discriminative space in particular, we
advocate the first approach.
