For Most Large Underdetermined Systems of Linear Equations the Minimal ` 1-norm Solution is also the Sparsest Solution

@inproceedings{Donoho2004ForML,
  title={For Most Large Underdetermined Systems of Linear Equations the Minimal ` 1-norm Solution is also the Sparsest Solution},
  author={David L. Donoho},
  year={2004}
}
We consider linear equations y = Φα where y is a given vector in R, Φ is a given n by m matrix with n < m ≤ An, and we wish to solve for α ∈ R. We suppose that the columns of Φ are normalized to unit ` norm 1 and we place uniform measure on such Φ. We prove the existence of ρ = ρ(A) 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 = Φα0 by a coefficient vector α0 ∈ R with fewer than ρ · n nonzeros, the… CONTINUE READING
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References

Publications referenced by this paper.
Showing 1-10 of 28 references

Matching pursuits with time-frequency dictionaries

IEEE Trans. Signal Processing • 1993
View 4 Excerpts
Highly Influenced

Greed is good: algorithmic results for sparse approximation

IEEE Transactions on Information Theory • 2004
View 7 Excerpts
Highly Influenced

On sparse representations in arbitrary redundant bases

IEEE Transactions on Information Theory • 2004
View 4 Excerpts
Highly Influenced

Optimally Sparse Representation from Overcomplete Dictionaries via `1 norm minimization

D. L. Donoho, Elad, Michael
Proc. Natl. Acad. Sci. USA March • 2002
View 6 Excerpts
Highly Influenced

Local Operator Theory, Random Matrices and Banach Spaces

S.J.K.R. Davidson
Szarek • 2001
View 2 Excerpts
Highly Influenced

Spaces with large distances to `∞ and random matrices

S. J. Szarek
Amer. Jour. Math. 112, • 1990
View 2 Excerpts
Highly Influenced

The Dimension of Almost Spherical Sections of Convex Bodies

J. LINDENSTRAUSS BY T. FIGIEL
1977
View 4 Excerpts
Highly Influenced

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