# Sparse nonnegative solution of underdetermined linear equations by linear programming.

```@article{Donoho2005SparseNS,
title={Sparse nonnegative solution of underdetermined linear equations by linear programming.},
author={David L. Donoho and Jared Tanner},
journal={Proceedings of the National Academy of Sciences of the United States of America},
year={2005},
volume={102 27},
pages={
9446-51
}
}```
• Published 2005
• Mathematics
• Proceedings of the National Academy of Sciences of the United States of America
Consider an underdetermined system of linear equations y = Ax with known y and d x n matrix A. We seek the nonnegative x with the 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. We explain this by the theory of convex polytopes. Let a(j) denote the jth column of A, 1 < or = j < or = n, let a0 = 0 and P denote the convex…
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