Necessary and sufficient conditions of solution uniqueness in ℓ1 minimization

—This paper shows that the solutions to various convex (cid:96) 1 minimization problems are unique if and only if a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as other (cid:96) 1 models that either minimize f ( Ax − b ) or impose the constraint f ( Ax − b ) ≤ σ , where f is a strictly convex function. For these models, this paper proves that, given a solution x ∗ and defining I = supp( x ∗ ) and s = sign( x ∗ I ) , x ∗ is the unique solution if and only if A I has full column rank and there exists y such that A TI y = s and | a Ti y | ∞ < 1 for i (cid:54)∈ I . This condition is previously known to be sufficient for the basis pursuit model to have a unique solution supported on I . Indeed, it is also necessary, and applies to a variety of other (cid:96) 1 models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically.