Babel function

Background

The Babel function measures the maximum total coherence between a fixed atom and a collection of other atoms in a dictionary.

Definition and Formulation

The Babel function (also known as cumulative coherence) 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 ${\boldsymbol {A}}$

References

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.84.5256&rep=rep1&type=pdf
http://www.yaroslavvb.com/papers/tropp-just.pdf