Babel function

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

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

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

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.^[3]

References

See also

• Compressed sensing

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