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 ${\displaystyle {\boldsymbol {A}}}$.[1][2]

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.[3]

References

1. Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation" (PDF). CiteSeerX 10.1.1.84.5256.
2. Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise
3. Huan Li and Zhouchen Lin. "Construction of Incoherent Dictionaries via Direct Babel Function Minimization" (PDF).

See also

• Compressed sensing