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 formulationEdit

The Babel function of a dictionary   with normalized columns is a real-valued function that is defined as


where   are the columns (atoms) of the dictionary  .[1][2]

Special caseEdit

When p=1, the babel function is the mutual coherence.

Pratical ApplicationsEdit

Li and Lin have used the Babel function to aid in creating effective dictionaries for Machine Learning applications.[3]


  1. ^ Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation" (PDF). IEEE Trans. Inform. Theory. 50 (10): 2231–2242. CiteSeerX doi:10.1109/TIT.2004.834793. S2CID 675692.
  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 alsoEdit