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 $a_1, \ldots, a_m\in {\mathbb C}^d$ 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

$M = \max_{1 \le i \ne j \le m} \left| a_i^H a_j \right|.$

A lower bound is

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

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

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.

See also

• Compressed sensing
• Restricted isometry property

Further reading

• Mutual coherence
• R1magic : R package providing mutual coherence computation.