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 [math]\displaystyle{ \boldsymbol{A} }[/math] with normalized columns is a real-valued function that is defined as
- [math]\displaystyle{ \mu_1(p) = \max_{ |\lambda| = p} \{ \max_{j\notin \lambda} \{ \sum_{i\in\lambda} {|\boldsymbol{a}_i^{\boldsymbol{T}}\boldsymbol{a}_j|} \} \} }[/math]
where [math]\displaystyle{ \boldsymbol{a}_k }[/math] are the columns (atoms) of the dictionary [math]\displaystyle{ \boldsymbol{A} }[/math].[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]
References
- ↑ Joel A. Tropp (2004). "Greed is good: Algorithmic results for sparse approximation". IEEE Trans. Inform. Theory 50 (10): 2231–2242. doi:10.1109/TIT.2004.834793. http://web.math.princeton.edu/tfbb/spring03/greed-ticam0304.pdf.
- ↑ 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". https://zhouchenlin.github.io/Publications/2018-ACML-Babel.pdf.
See also
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