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<p/> <p>Given a signal <inline-formula><graphic file="1687-6180-2006-096247-i1.gif"/></inline-formula> and a full-rank matrix <inline-formula><graphic file="1687-6180-2006-096247-i2.gif"/></inline-formula> with <inline-formula><graphic file="1687-6180-2006-096247-i3.gif"/></inline-formula>, we define the signal's overcomplete representations as all <inline-formula><graphic file="1687-6180-2006-096247-i4.gif"/></inline-formula> satisfying <inline-formula><graphic file="1687-6180-2006-096247-i5.gif"/></inline-formula>. Among all the possible solutions, we have special interest in the sparsest one&#8212;the one minimizing <inline-formula><graphic file="1687-6180-2006-096247-i6.gif"/></inline-formula>. Previous work has established that a representation is unique if it is sparse enough, requiring <inline-formula><graphic file="1687-6180-2006-096247-i7.gif"/></inline-formula>. The measure <inline-formula><graphic file="1687-6180-2006-096247-i8.gif"/></inline-formula> stands for the minimal number of columns from <inline-formula><graphic file="1687-6180-2006-096247-i9.gif"/></inline-formula> that are linearly dependent. This bound is tight&#8212;examples can be constructed to show that with <inline-formula><graphic file="1687-6180-2006-096247-i10.gif"/></inline-formula> or more nonzero entries, uniqueness is violated. In this paper we study the behavior of overcomplete representations beyond the above bound. While tight from a worst-case standpoint, a probabilistic point of view leads to uniqueness of representations satisfying <inline-formula><graphic file="1687-6180-2006-096247-i11.gif"/></inline-formula>. Furthermore, we show that even beyond this point, uniqueness can still be claimed with high confidence. This new result is important for the study of the average performance of pursuit algorithms&#8212;when trying to show an equivalence between the pursuit result and the ideal solution, one must also guarantee that the ideal result is indeed the sparsest.</p

Sparse Representations Are Most Likely to Be the Sparsest Possible