Mutual Coherence, Spark and Babel Function

Mutual Coherence and Babel Function are the properties of a matrix, used to estimate the Spark of a matrix, which in turn is used to determine the optimality of the solution to \(\text{P}_0\) problem.

Babel Function gives a tighter bound on the Spark of a matrix.

Spark of a matrix \(\boldsymbol{A}\) is the size of the smallest subset of linearly dependent columns of \(\boldsymbol{A}\).

mutual_coherence(mat)	For an arbitrary input matrix \(\boldsymbol{A}\) of size N x M, the mutual coherence is the maximal absolute inner-product between its normalized columns \(\{ a_i \mid i=1,2,...,M \}\):
babel(mat)	For an arbitrary input matrix \(\boldsymbol{A}\) of size N x M and normalized columns \(\{ a_i \mid i=1,2,...,M \}\), the Babel-Function is defined by