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.

See also

• Compressed sensing