Algorithms for sparse optimization : complexity and applications
稀疏優化演算法 : 複雜度與應用
Student thesis: Doctoral Thesis
Related Research Unit(s)
Exploiting the sparsity of an underlying problem is an important research topic that has extensive applications in statistics, machine learning, signal processing, compressed sensing and so on. One method by which to exploit sparsity is to add a sparse regularizer, such as L0 norm, into the original model. However, such sparse models are often limited by high computational cost attributed to the combinatorial property of the sparse regularizer. This property limits the application of sparse models to small-sized problems. The recently developed compressed sensing theory shows that the convex relaxation of some sparse models has the same solution as that of the original model under certain conditions. This situation enables the design of efficient algorithms to solve large-scale sparse problems. In the same vein, a class of non-convex but continuous sparse models has been proposed. Non-convex sparse models not only possess a number of desirable properties relative to variable selection, but are also able to obtain sparse solutions under weaker conditions than the convex relaxation. In this dissertation, we systematically study on a class of non-convex continuous sparse models and establish efficient algorithms to solve the related problems. In Chapter 2, we discuss the properties of a class of non-convex continuous sparse models (L2 − Lp model) in terms of optimization. In particular, we present the characteristics of the optimal solutions, discuss the relationships between solution sparsity and model parameters, as well as examine the connections among several sparse models. These results provide a foundation for subsequently developing efficient algorithms. We have to point out that several results from other researchers are also included for completeness of our discussion in Chapter 2. Chapter 3 presents a fast first-order iterative algorithm for solving linear constrained non-convex minimization problems. We show that this algorithm can obtain an E-KKT point within O(ϵ−2) iterations. This algorithm is extended to solve the L2 − Lp minimization problem with the smoothing and lower bound cut techniques to deal with non-Lipschitz hardness of the objective function. The complexity result still holds for the linear constrained L2 − Lp minimization problem. We like-wise use several techniques, such as noise debiasing and adaptive mechanism for parameter selection to improve algorithm speed and solution quality. Numerical results on several applications demonstrate the efficiency of the proposed algorithms. In Chapter 4, we subsequently propose a breakthrough linear convergence iterative algorithm to find an ϵ-KKT solution for the linear constrained L2 − Lp minimization problem under mild conditions. By analyzing the special structure of the objective function, we use several interesting techniques to separate the convex, concave and non-Lipschitz continuous situations, and handle them differently. Our analysis shows that the proposed algorithm has a linear convergence rate for obtaining an ϵ-KKT solution, thus demonstrating a significant improvement over the current best result O(ϵ-3/2). Lastly, we discuss future research directions.
- Mathematical optimization, Sparse matrices.