Babel function

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

${\displaystyle \mu _{1}(p)=\max _{|\lambda |=p}\{\max _{j\notin \lambda }\{\sum _{i\in \lambda }{|{\boldsymbol {a}}_{i}^{\boldsymbol {T}}{\boldsymbol {a}}_{j}|}\}\}}$

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

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

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

• Compressed sensing