Mutual coherence (linear algebra)

In linear algebra, the coherence or mutual coherence of a matrix A is defined as the maximum absolute value of the cross-correlations between the columns of A.[1][2]

Formally, let ${\displaystyle a_{1},\ldots ,a_{m}\in {\mathbb {C} }^{d}}$ be the columns of the matrix A, which are assumed to be normalized such that ${\displaystyle a_{i}^{H}a_{i}=1.}$ The mutual coherence of A is then defined as[1][2]

${\displaystyle M=\max _{1\leq i\neq j\leq m}\left|a_{i}^{H}a_{j}\right|.}$

A lower bound is [3]

${\displaystyle M\geq {\sqrt {\frac {m-d}{d(m-1)}}}}$

A deterministic matrix with the mutual coherence almost meeting the lower bound can be constructed by Weil's theorem.[4]

This concept was reintroduced by David Donoho and Michael Elad in the context of sparse representations.[5] A special case of this definition for the two-ortho case appeared earlier in the paper by Donoho and Huo.[6] The mutual coherence has since been used extensively in the field of sparse representations of signals. In particular, it is used as a measure of the ability of suboptimal algorithms such as matching pursuit and basis pursuit to correctly identify the true representation of a sparse signal.[1][2][7] Joel Tropp introduced a useful extension of Mutual Coherence, known as the Babel function, which extends the idea of cross-correlation between pairs of columns to the cross-correlation from one column to a set of other columns. The Babel function for two columns is exactly the Mutual coherence, but it also extends the coherence relationship concept in a way that is useful and relevant for any number of columns in the sparse representation matix as well.[8]