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

Abstract

We consider linear equations y = Φx where y is a given vector in ℝⁿ and Φ is a given n × m matrix with n < m ≤ τn, and we wish to solve for x ∈ ℝ^m. We suppose that the columns of Φ are normalized to the unit ��₂-norm, and we place uniform measure on such Φ. We prove the existence of ρ = ρ(τ) > 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 ∈ ℝ^mwith fewer than ρ · n nonzeros, the solution x₁of the ��₁-minimization problem

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is unique and equal to x₀. In contrast, heuristic attempts to sparsely solve such systems—greedy algorithms and thresholding—perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost-spherical sections in Banach space theory, and deviation bounds for the eigenvalues of random Wishart matrices. © 2006 Wiley Periodicals, Inc.