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 ℝnand Φ is a given n×m matrix with n<m≤ τ n, and we wish to solve for x∈ ℝm. We suppose that the columns of Φ are normalized to the unit 𝓁 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= Φ x0by a coefficient vector x0∈ ℝmwith fewer than ρ · n nonzeros, the solution x1of the 𝓁 1‐minimization problemis 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.