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 \le i \ne j \le m} \left| a_i^H a_j \right|.

A lower bound is [3]

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.[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

References

  1. ↑ 1.0 1.1 1.2 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. ↑ 2.0 2.1 2.2 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