For Most Large Underdetermined Systems of Equations , the Minimal ` 1-norm Near-Solution Approximates the Sparsest Near-Solution

@inproceedings{Donoho2004ForML,
  title={For Most Large Underdetermined Systems of Equations , the Minimal ` 1-norm Near-Solution Approximates the Sparsest Near-Solution},
  author={David L. Donoho},
  year={2004}
}
We consider inexact linear equations y ≈ Φα where y is a given vector in R, Φ is a given n by m matrix, and we wish to find an α0, which is sparse and gives an approximate solution, obeying ‖y − Φα0, ‖2 ≤ . In general this requires combinatorial optimization and so is considered intractable. On the other hand, the ` minimization problem min ‖α‖1 subject to ‖y − Φα‖2 ≤ , is convex, and is considered tractable. We show that for most Φ the solution α̂1, = α̂1, (y, Φ) of this problem is quite… CONTINUE READING
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