  This paper studies the question of how well a signal can be reprsented by a
sparse linear combination of reference signals from an overcomplete dictionary.
When the dictionary size is exponential in the dimension of signal, then the
exact characterization of the optimal distortion is given as a function of the
dictionary size exponent and the number of reference signals for the linear
representation. Roughly speaking, every signal is sparse if the dictionary size
is exponentially large, no matter how small the exponent is. Furthermore, an
iterative method similar to matching pursuit that successively finds the best
reference signal at each stage gives asymptotically optimal representations.
This method is essentially equivalent to successive refinement for multiple
descriptions and provides a simple alternative proof of the successive
refinability of white Gaussian sources.
