Mutual coherence (linear algebra)

In linear algebra, the coherence^[1] or mutual coherence^[2] 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^[1][2]

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

A lower bound is ^[3]

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

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

The concept was introduced in a slightly less general framework by David Donoho and Xiaoming Huo,^[5] 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.^[1][2][6]

See also

• Compressed sensing
• Restricted isometry property

References

1. Tropp, J.A. (March 2006). "Just relax: Convex programming methods for identifying sparse signals in noise". IEEE Transactions on Information Theory. 52 (3): 1030–1051. doi:10.1109/TIT.2005.864420.
2. Donoho, D.L.; M. Elad; V.N. Temlyakov (January 2006). "Stable recovery of sparse overcomplete representations in the presence of noise". IEEE Transactions on Information Theory. 52 (1): 6–18. doi:10.1109/TIT.2005.860430.
3. Welch, L. R. (1974). "Lower bounds on the maximum cross-correlation of signals". IEEE Transactions on Information Theory. 20: 397–399. doi:10.1109/tit.1974.1055219.
4. Zhiqiang, Xu (April 2011). "Deterministic Sampling of Sparse Trigonometric Polynomials". Journal of Complexity. 27 (2): 133–140. doi:10.1016/j.jco.2011.01.007.
5. Donoho, D.L.; Xiaoming Huo (November 2001). "Uncertainty principles and ideal atomic decomposition". IEEE Transactions on Information Theory. 47 (7): 2845–2862. doi:10.1109/18.959265.
6. Fuchs, J.-J. (June 2004). "On sparse representations in arbitrary redundant bases". IEEE Transactions on Information Theory. 50 (6): 1341–1344. doi:10.1109/TIT.2004.828141.

Further reading

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