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sparse representation is:
A signal can be represented as a linear combination of basis functions where the set of basis functions is called dictionary and data samples are much more than their features. Mathematically, in the system of linear equations $Y=DX$ where $Y \in \mathbb{R}^{n \times N} (n \ll N)$ we seek a dictionary that results in sparse representation of $Y$.

Digging the history of sparse representation, my take is it all started by using transforms as D; wavelet, Fourier and such. They were good in the application they were designed for but a generalization to a wider area was a failure. So a combination of basis functions was used as a dictionary D, an overcomplete one with more basis functions than the size of each basis function. There are lots of examples of different transforms and their result. But I cannot think of an example for an overcomplete dictionary. Is it due to the fact that they are learned and there is no special dictionary to be used? I mean, given an initial dictionary, it is learned in an iterative process to better fit the data, is there a well-known instance of such a dictionary like the case JPEG2000 using wavelet packets as D?

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