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.

Formally, let a 1 , … , a m ∈ C d be the columns of the matrix A, which are assumed to be normalized such that a i H a i = 1. The mutual coherence of A is then defined as

A lower bound is

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

The concept was introduced in a slightly less general framework by David Donoho and Xiaoming Huo, and 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.