We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed l_1 (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates l_1 and l_1 norms through a nonnegative parameter l_1 , similar to l_1 with l_1 , and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem and discuss the exact recovery of l_1 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l_1 norm minimal solution if the sensing matrix l_1 satisfies a restricted isometry property (RIP). Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The inner loop concerns an l_1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value l_1 , and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l_1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix l_1 . DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two

We study analytical and numerical properties of the L 1 - L 2 minimization problem for sparse representation of a signal over a highly coherent dictionary. Though the L 1 - L 2 metric is non-convex, it is Lipschitz continuous. The difference of convex algorithm (DCA) is readily applicable for computing the sparse representation coefficients. The L 1 - L 2 minimization appears as an initialization step of DCA. We further integrate DCA with a non-standard simulated annealing methodology to approximate globally sparse solutions. Non-Gaussian random perturbations are more effective than standard Gaussian perturbations for improving sparsity of solutions. In numerical experiments, we conduct an extensive comparison among sparse penalties such as L 1 - L 2 for L 1 - L 2 based on data from three specific applications (over-sampled discreet cosine basis, differential absorption optical spectroscopy, and image denoising) where

Unmixing problems in many areas such as hyperspectral imaging and differential optical absorption spectroscopy (DOAS) often require finding sparse nonnegative linear combinations of dictionary elements that match observed data. We show how aspects of these problems, such as misalignment of DOAS references and uncertainty in hyperspectral endmembers, can be modeled by expanding the dictionary with grouped elements and imposing a structured sparsity assumption that the combinations within each group should be sparse or even 1-sparse. If the dictionary is highly coherent, it is difficult to obtain good solutions using convex or greedy methods, such as nonnegative least squares (NNLS) or orthogonal matching pursuit. We use penalties related to the Hoyer measure, which is the ratio of the l_1 and l_1 norms, as sparsity penalties to be added to the objective in NNLS-type models. For solving the

The goal of smooth function reconstruction on a 2D or 3D manifold is to obtain a smooth function on surfaces or higher dimensional manifolds. It is a common problem in computer graphics and computational mathematics, especially in civil engineering including structural analysis of solid objects. In this chapter, we introduce a new method using harmonic functions for solving this problem. This method contains the following steps: (1) Partition the boundary surfaces of the 3D manifold based on sample points so that each sample point is on the edge of the partition. (2) Use gradually varied interpolation on the edges so that each point on the edge will be assigned a value. In addition, all values on the edge are gradually varied. (3) Use discrete harmonic functions to fit the unknown points, i.e. the points inside each partition patch. This solution of the fitting becomes the piecewise harmonic function.

In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. We invented numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow. We also showed broad applications using discrete Ricci flow in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces.