## Sparse nonnegative solutions of underdetermined linear equations by linear programming (2005)

Venue: | Proceedings of the National Academy of Sciences |

Citations: | 106 - 6 self |

### BibTeX

@INPROCEEDINGS{Donoho05sparsenonnegative,

author = {David L. Donoho and Jared Tanner},

title = {Sparse nonnegative solutions of underdetermined linear equations by linear programming},

booktitle = {Proceedings of the National Academy of Sciences},

year = {2005},

pages = {9446--9451}

}

### Years of Citing Articles

### OpenURL

### Abstract

Consider an underdetermined system of linear equations y = Ax with known d×n matrix A and known y. We seek the sparsest nonnegative solution, i.e. the nonnegative x with fewest nonzeros satisfying y = Ax. In general this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. In classical convex polytope theory, a polytope P is called k-neighborly if every set of k vertices of P span a face of P. Let aj denote the j-th column of A, 1 ≤ j ≤ n, let a0 = 0 and let P denote the convex hull of the aj. We say P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is completely equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. Using this and classical results on polytope neighborliness we obtain two types of corollaries. First, because many ⌊d/2⌋-neighborly polytopes are known, there are many systems where the sparsest solution is available by convex optimization rather than combinatorial