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In this paper, we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell _0$ pseudo norm is able to better induce sparsity than the commonly used $\ell _1$ norm. For a class of weakly convex sparsity inducing functions, we prove the nonconvexity of the corresponding problem and study its local optimality conditions and the choice of the regularization parameter. Despite the nonconvexity, a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression, and its convergence behavior is studied theoretically. Then, the general framework is applied to a specific weakly convex function, and a local optimality condition and a bound on the logistic loss at a local optimum are provided. The solution method is instantiated in this case as an iterative firm-shrinkage algorithm, and a Nesterov acceleration is used with a convergence guarantee. Its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets. [ABSTRACT FROM AUTHOR]

The construction of highly incoherent frames, sequences of vectors placed on the unit hyper sphere of a finite dimensional Hilbert space with low correlation between them, has proven very difficult. Algorithms proposed in the past have focused on minimizing the absolute value off-diagonal entries of the Gram matrix of these structures. Recently, a method based on convex optimization that operates directly on the vectors of the frame has been shown to produce promising results. This paper gives a detailed analysis of the optimization problem at the heart of this approach and, based on these insights, proposes a new method that substantially outperforms the initial approach and all current methods in the literature for all types of frames, with low and high redundancy. We give extensive experimental results that show the effectiveness of the proposed method and its application to optimized compressed sensing. [ABSTRACT FROM PUBLISHER]

CG_DESCENT is a state-of-the-art algorithm to solve large-scale unconstrained minimization problems. However, research activities on CG_DESCENT in some other scenarios are relatively fewer. In this paper, by combining with the projection method of Solodov and Svaiter, we extend CG_DESCENT to solve large-scale nonlinear convex constrained monotone equations. The proposed method does not require the Jacobian information, even though it does not store any matrix at each iteration. It thus has the potential to solve large-scale non-smooth problems. Under some mild conditions, we show that the proposed method converges globally. Primary numerical results illustrate that the proposed method works quite well. Moreover, we also extend this method to solve the -norm regularized problems to decode a sparse signal in compressive sensing. Performance comparisons show that the proposed method is practical, efficient and competitive with the compared ones. [ABSTRACT FROM AUTHOR]