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.

Definition and formulation[edit]

The Babel function of a dictionary \boldsymbol{A} with normalized columns is a real-valued function that is defined as

\mu(p) = \max_{ |\lambda| = p} \{ \max_{j\notin \lambda} \{   \sum_{i\in\lambda}  {|\boldsymbol{a}_i^{\boldsymbol{T}}\boldsymbol{a}_j|} \} \}

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

Special case[edit]

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

References[edit]

  1. ^ Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation". CiteSeerX: 10.1.1.84.5256. 
  2. ^ Just Relax: Convex Programming Methods for Identifying Sparse Signals in Noise

See also[edit]