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

Source abstract

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 < m ≤ τ n , and we wish to solve for x ∈ ℝ m . 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 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.