Babel function explained

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 \{ max _{j\notin} \{ \sum _{i\inλ} {|\boldsymbol{a} ^{\boldsymbol{T} _{i} }\boldsymbol_j|} \} \}where \boldsymbol{a} _{k} are the columns (atoms) of the dictionary \boldsymbol{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]

See also

• Compressed sensing

Notes and References

1. Greed is good: Algorithmic results for sparse approximation . Joel A. Tropp . IEEE Trans. Inform. Theory . 2004 . 50 . 10 . 2231–2242 . 10.1109/TIT.2004.834793 . 10.1.1.84.5256 . 675692 .
2. http://www.yaroslavvb.com/papers/tropp-just.pdf Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise
3. Web site: Construction of Incoherent Dictionaries via Direct Babel Function Minimization. Huan Li and Zhouchen Lin .