Minimizing the l0-seminorm of a vector under convex constraints is a combinatorial (NP-hard) problem. Replacement of the l0seminorm with the l1-norm is a commonly used approach to compute an approximate solution of the original l0-minimization problem by means of convex programming. In the theory of compressive sensing, the condition that the sensing matrix satisfies the Restricted Isometry Pro...

Let l = [l0, l1] be the directed line segment from l0 ∈ IR to l1 ∈ IR. Suppose l̄ = [l̄0, l̄1] is a second segment of equal length such that l, l̄ satisfy the “two sticks condition”: ∥∥l1 − l̄0∥∥ ≥ ‖l1 − l0‖ ,∥∥l̄1 − l0∥∥ ≥ ∥∥l̄1 − l̄0∥∥ . Here ‖·‖ is a norm on IR. We explore the manner in which l1− l̄1 is then constrained when assumptions are made about “intermediate points” l∗ ∈ l, l̄∗ ∈ l̄. Roughly spe...

smooth coefficients aα(x), L+ be a formally adjoint differential operation. Let L0, L0 be minimal operators (i.e., for example, D(L0) is the clozure of C∞ 0 (Ω) in the norm of the graph ‖u‖L = ‖u‖L2(Ω)+‖Lu‖L2(Ω)), and L, L+ be maximal expansions of L,L+ in the space L2(Ω) respectively (i.e. L = (L0 ) ∗, L+ = (L0)∗), L̃ = L|D(L̃) where D(L̃) is the clozure of C∞(Ω̄) in the norm of the graph ‖u‖L and...

Finding the sparsest, or minimum l0-norm, representation of a signal given an overcomplete dictionary of basis vectors is an important problem in many application domains. Unfortunately, the required optimization problem is often intractable because there is a combinatorial increase in the number of local minima as the number of candidate basis vectors increases. This deficiency has prompted mo...

We present an anisotropic point cloud denoising method using L0 minimization. The L0 norm directly measures the sparsity of a solution, and we observe that many common objects can be defined as piece-wise smooth surfaces with a small number of features. Hence, we demonstrate how to apply an L0 optimization directly to point clouds, which produces sparser solutions and sharper surfaces than eith...

In this paper, we have proved that in every underdetermined linear system Ax = b there corresponds a constant p∗(A, b) > 0 such that every solution to the lp-norm minimization problem also solves the l0-norm minimization problem whenever 0 < p < p∗(A, b). This phenomenon is named NP/CMP equivalence.

With the previously proposed non-uniform norm called lN -norm, which consists of a sequence of l1-norm or l0-norm elements according to relative magnitude, a novel lN-norm sparse recovery algorithm can be derived by projecting the gradient descent solution to the reconstruction feasible set. In order to gain analytical insights into the performance of this algorithm, in this letter we analyze t...

It is proved in this paper that to every underdetermined linear system Ax = b there corresponds a constant p(A, b) > 0 such that every solution to the lp-norm minimization problem also solves the l0-norm minimization problem whenever 0 < p < p(A, b). This phenomenon is named NP/CLP equivalence.