Babel function

The Babel function (also known as cumulative coherence) measures the maximum total coherence between a fixed atom and a collection of other atoms in a dictionary. The Babel function was conceived of in the context of signals for which there exists a sparse representation consisting of atoms or columns of a redundant dictionary matrix, A.

Definition and formulation
The Babel function of a dictionary $$\boldsymbol{A}$$ with normalized columns is a real-valued function that is defined as
 * $$\mu_1(p) = \max_{ |\lambda| = p} \{ \max_{j\notin \lambda} \{  \sum_{i\in\lambda}  {|\boldsymbol{a}_i^{\boldsymbol{T}}\boldsymbol{a}_j|} \} \}$$

where $$\boldsymbol{a}_k$$ are the columns (atoms) of the dictionary $$\boldsymbol{A} $$.

Special case
When p=1, the Babel function is the mutual coherence.

Practical Applications
Li and Lin have used the Babel function to aid in creating effective dictionaries for machine learning applications.