For Most Large Underdetermined Systems of Linear Equations the Minimal `-norm Solution is also the Sparsest Solution

We consider linear equations y = Φα where y is a given vector in R, Φ is a given n by m matrix with n < m ≤ An, and we wish to solve for α ∈ R. We suppose that the columns of Φ are normalized to unit ` norm 1 and we place uniform measure on such Φ. We prove the existence of ρ = ρ(A) 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 = Φα0 by a coefficient vector α0 ∈ R with fewer than ρ · n nonzeros, the solution α1 of the ` minimization problem min ‖x‖1 subject to Φα = y is unique and equal to α0. 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.