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 with normalized columns is a real-valued function that is defined as
where are the columns (atoms) of the dictionary .[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" (PDF). CiteSeerX 10.1.1.84.5256.
- ^ 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" (PDF).