For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution

@article{Donoho2006ForML,
  title={For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution},
  author={D. Donoho},
  journal={Communications on Pure and Applied Mathematics},
  year={2006},
  volume={59},
  pages={797-829}
}
  • D. Donoho
  • Published 2006
  • Mathematics
  • Communications on Pure and Applied Mathematics
  • We consider linear equations y = Φx where y is a given vector in ℝn and Φ is a given n × m matrix with n 0 so that for large n and for all Φ's except a negligible fraction, the following property holds: For every y having a representation y = Φx0by a coefficient vector x0 ∈ ℝmwith fewer than ρ · n nonzeros, the solution x1of the 1-minimization problem is unique and equal to x0. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform… CONTINUE READING
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