Almost Chebyshev properties for L1-approximation of continuous functions
Journal of Approximation Theory
Linear L1 Approximation for a Discrete Point Set and L1 Solutions of Overdetermined Linear Equations
Journal of the ACM (JACM)
Efficient Minimization Methods of Mixed l2-l1 and l1-l1 Norms for Image Restoration
SIAM Journal on Scientific Computing
Sparse representations and approximation theory
Journal of Approximation Theory
Alternating Direction Algorithms for $\ell_1$-Problems in Compressive Sensing
SIAM Journal on Scientific Computing
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A real m × n matrix A and a vector y ∈ Rm determine the discrete l 1-regularization (DLR) problem 0.1 $$ \min \left\{\mbox{\,}|y-Ax|_1+\rho |x|_1:\,x\in\mathbb{R}^n \right\}, $$ where | · |1 denotes the l 1-norm of a vector and ρ ≥ 0 is a nonnegative parameter. In this paper, we provide a detailed analysis of this problem which include a characterization of all solutions to (0.1), remarks about the geometry of the solution set and an effective iterative algorithm for numerical solution of (0.1). We are specially interested in the behavior of the solution of (0.1) as a function of ρ and in this regard, we prove in general the existence of critical values of ρ between which the l 1-norm of any solution remains constant. These general remarks are significantly refined when A is a strictly totally positive (STP) matrix. The importance of STP matrices is well-established [5, 14]. Under this setting, the relationship between the number of nonzero coordinates of a distinguished solution of the DLR problem is precisely explained as a function of the regularization parameter for a certain class of vectors in Rm. Throughout our analysis of the DLR problem, we emphasize the importance of the dual maximum problem by demonstrating that any solution of it leads to a solution of the DLR problem, and vice versa.