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Babel function

From Wikipedia, the free encyclopedia

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

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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

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

Practical Applications

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

References

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  1. ^ Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation" (PDF). IEEE Trans. Inform. Theory. 50 (10): 2231–2242. CiteSeerX 10.1.1.84.5256. 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 also

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