spark of a matrix
Let be a field, and let Consider an matrix whose entries belong to Let be the columns of
The spark of the matrix is defined to be the infimum of the set of all positive integers with the property that
The notion of spark of a matrix was introduced by Donoho and Elad in . It is strictly related to Compressed Sensing. The word “spark” comes from a verbal fusion of “sparse” and “rank”.
The following properties hold true:
(i) (ii) (iii) if and only if the matrix has a zero column, (iv) if then
Moreover, coincides with the infimum of the set , where is the Hamming weight of and stands for the appropriate zero vectors.
- 1 D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via minimization, Proc. Natl. Acad. Sci. USA 100, No. 5: 2197 – 2202 (2003).
|Title||spark of a matrix|
|Date of creation||2013-03-22 19:36:23|
|Last modified on||2013-03-22 19:36:23|
|Last modified by||kammerer (26336)|