# 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 [math]\displaystyle{ \boldsymbol{A} }[/math] with normalized columns is a real-valued function that is defined as

- [math]\displaystyle{ \mu_1(p) = \max_{ |\lambda| = p} \{ \max_{j\notin \lambda} \{ \sum_{i\in\lambda} {|\boldsymbol{a}_i^{\boldsymbol{T}}\boldsymbol{a}_j|} \} \} }[/math]

where [math]\displaystyle{ \boldsymbol{a}_k }[/math] are the columns (atoms) of the dictionary [math]\displaystyle{ \boldsymbol{A} }[/math].^{[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

- ↑ Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation".
*IEEE Trans. Inform. Theory***50**(10): 2231–2242. doi:10.1109/TIT.2004.834793. http://web.math.princeton.edu/tfbb/spring03/greed-ticam0304.pdf. - ↑ Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise
- ↑ Huan Li and Zhouchen Lin. "Construction of Incoherent Dictionaries via Direct Babel Function Minimization". https://zhouchenlin.github.io/Publications/2018-ACML-Babel.pdf.

## See also

Original source: https://en.wikipedia.org/wiki/Babel function.
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