For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution

Abstract We consider linear equations y = Φ x where y is a given vector in ℝ n 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 = Φ x 0 by a coefficient vector x 0 ∈ ℝ m with fewer than ρ · n nonzeros, the solution x 1 of the 𝓁 1 ‐minimization problem is unique and equal to x 0 . 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.