# 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.

## Definition and formulation

The Babel function of a dictionary $\boldsymbol{A}$ with normalized columns is a real-valued function that is defined as

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

where $\boldsymbol{a}_k$ are the columns (atoms) of the dictionary $\boldsymbol{A}$.[1][2]

## Special case

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

## References

1. ^ Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation". CiteSeerX: 10.1.1.84.5256.
2. ^ Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise