# 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 $\displaystyle{ \boldsymbol{A} }$ with normalized columns is a real-valued function that is defined as

$\displaystyle{ \mu_1(p) = \max_{ |\lambda| = p} \{ \max_{j\notin \lambda} \{ \sum_{i\in\lambda} {|\boldsymbol{a}_i^{\boldsymbol{T}}\boldsymbol{a}_j|} \} \} }$

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

## Special case

When p=1, the babel function is the mutual coherence.

## Pratical Applications

Li and Lin have used the Babel function to aid in creating effective dictionaries for Machine Learning applications.