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 be the columns of the matrix A, which are assumed to be normalized such that The mutual coherence of A is then defined as[1][2]

A lower bound is [3]

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

This concept was introduced by David Donoho and Michael Elad.[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]

See also

References

  1. 1 2 3 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. 1 2 3 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&ndash, 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.; Michael Elad (March 2003). "Optimally sparse representation in general (nonorthogonal) dictionaries via L1 minimization". Proc. Natl. Acad. Sci. 100: 2197&ndash, 2202. doi:10.1073/pnas.0437847100. PMC 153464.
  6. ↑ Donoho, D.L.; Xiaoming Huo (November 2001). "Uncertainty principles and ideal atomic decomposition". IEEE Transactions on Information Theory. 47 (7): 2845&ndash, 2862. doi:10.1109/18.959265.
  7. ↑ Fuchs, J.-J. (June 2004). "On sparse representations in arbitrary redundant bases". IEEE Transactions on Information Theory. 50 (6): 1341&ndash, 1344. doi:10.1109/TIT.2004.828141.

Further reading


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