Designing computational experiments involving ℓ1 minimization with linear constraints in a finite-dimensional, real-valued space for receiving a sparse solution with a precise number k of nonzero entries is, in general, difficult. Several conditions were introduced which guarantee that, for example for small k or for certain matrices, simply placing entries with desired characteristics on a randomly chosen support will produce vectors which can be recovered by ℓ1 minimization. In this work, we consider the case of large k and introduce a method which constructs vectors which support has the cardinality k and which can be recovered via ℓ1 minimization. Especially, such vectors with largest possible support can be constructed. Further, we propose a methodology to quickly check whether a given vector is recoverable. This method can be cast as a linear program and we compare it with solving ℓ1 minimization directly. Moreover, we gain new insights in the recoverability in a non-asymptotic regime. Our proposal for quickly checking vectors bases on optimality conditions for exact solutions of the ℓ1 minimization. These conditions can be used to establish equivalence classes of recoverable vectors which have a support of the same cardinality. Further, by these conditions we deduce a geometrical interpretation which identifies an equivalence class with a face of an hypercube which is cut by a certain affine subspace. Due to the new geometrical interpretation we derive new results on the number of equivalence classes which are illustrated by computational experiments.
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