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 $𝑨$ with normalized columns is a real-valued function that is defined as

${\mu }_1(p)=\underset{|\lambda |=p}{\max }\{\underset{j\notin \lambda }{\max }\{\sum _{i\in \lambda }|{𝒂}_i^{𝑻}{𝒂}_j|\}\}$

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

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