We investigate the recovery of almost s-sparse vectors x ∈ C N from undersampled and inaccurate data y = Ax + e ∈ C m by means of minimizing z 1 subject to the equality constraints Az = y. If m s ln(N/s) and if Gaussian random matrices A ∈ R m×N are used, this equality-constrained 1 -minimization is known to be stable with respect to sparsity defects and robust with respect to measurement errors. If m s ln(N/s) and if Weibull random matrices are used, we prove here that the equality-constrained 1 -minimization remains stable and robust. The arguments are based on two key ingredients, namely the robust null space property and the quotient property. The robust null space property relies on a variant of the classical restricted isometry property where the inner norm is replaced by the 1 -norm and the outer norm is replaced by a norm comparable to the 2 -norm. For the 1 -minimization subject to inequality constraints, this yields stability and robustness results that are also valid when considering sparsity relative to a redundant dictionary. As for the quotient property, it relies on lower estimates for the tail probability of sums of independent Weibull random variables. This article deals with the approximate recovery of signals using 1 -minimizations. In classical works, the signals f are assumed to be compressible in an orthonormal basis, and they are identified with their vectors x ∈ C N of coefficients on this orthonormal basis. The signals are then acquired via the measurement vectors y = Ax + e ∈ C m , m N , where A ∈ C m×N is a measurement matrix (the same for all signals) and e ∈ C m are error vectors. If the measurement matrix A satisfies a restricted isometry condition of order s and if an estimate e 2 ≤ η is available, then the reconstruction map given by the inequality-constrained 1 -minimization ∆ 1,η (y) := argmin z∈C N z 1 subject to Az − y 2 ≤ η yields bounds on the reconstruction error of the type x − ∆ 1,η (Ax + e) 2 ≤ C √ s σ s (x) 1 + D η, (2) *
The analysis in REF incorporates the robust null space property, the verification of which for the Weibull matrices relies on a variant of the classical restricted isometry property.
Stability and robustness of ℓ1-minimizations with Weibull matrices and redundant dictionaries
Linear Algebra and its Applications
Mathematics