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

• Published in: Communications on pure and applied mathematics Vol. 59; no. 6; pp. 797 - 829
• Main Author: Donoho, David L.
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.06.2006
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20132

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 ∈ ℝm. We suppose that the columns of Φ are normalized to the unit 𝓁2‐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 = Φx0 by a coefficient vector x0 ∈ ℝm with fewer than ρ · n nonzeros, the solution x1 of the 𝓁1‐minimization problem $${\rm min} \|x\|_{1} \;\;{subject \; to}\;\; \Phi x = y$$ 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.