In linear algebra

Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, the coherence or mutual coherence of a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

 A is defined as the maximum absolute value of the cross-correlation

Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long-duration signal for a shorter, known feature...

s 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

The concept was introduced in a slightly less general framework by Donoho

David Donoho

David Leigh Donoho, born on March 5, 1957 in Los Angeles, is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences...

 and Huo, and has since been used extensively in the field of sparse representations

Sparse approximation

Sparse approximation is the problem of finding a signal or vector estimate with sparseness property, that is having a small number of nonzero elements, that satisfies a system of equations....

 of signal

Signal (electrical engineering)

In the fields of communications, signal processing, and in electrical engineering more generally, a signal is any time-varying or spatial-varying quantity....

s. In particular, it is used as a measure of the ability of suboptimal algorithms such as matching pursuit

Matching pursuit

Matching pursuit is a type of numerical technique which involves finding the "best matching" projections of multidimensional data onto an over-complete dictionary D...

 and basis pursuit to correctly identify the true representation of a sparse signal.