Optimal k-thresholding algorithms for sparse optimization problems

The simulations indicate that the existing hard thresholding technique independent of the residual function may cause a dramatic increase or numerical oscillation of the residual. This inherit drawback of the hard thresholding renders the traditional thresholding algorithms unstable and thus generally inefficient for solving practical sparse optimization problems. How to overcome this weakness and develop a truly efficient thresholding method is a fundamental question in this field. The aim of this paper is to address this question by proposing a new thresholding technique based on the notion of optimal $k$-thresholding. The central idea for this new development is to connect the $k$-thresholding directly to the residual reduction during the course of algorithms. This leads to a natural design principle for the efficient thresholding methods. Under the restricted isometry property (RIP), we prove that the optimal thresholding based algorithms are globally convergent to the solution of sparse optimization problems. The numerical experiments demonstrate that when solving sparse optimization problems, the traditional hard thresholding methods have been significantly transcended by the proposed algorithms which can even outperform the classic $\ell_1$-minimization method in many situations

A Unified Approach to Synchronization Problems over Subgroups of the Orthogonal Group

Given a group $\mathcal{G}$, the problem of synchronization over the group $\mathcal{G}$ is a constrained estimation problem where a collection of group elements $G^*_1, \dots, G^*_n \in \mathcal{G}$ are estimated based on noisy observations of pairwise ratios $G^*_i {G^*_j}^{-1}$ for an incomplete set of index pairs $(i,j)$. This problem has gained much attention recently and finds lots of applications due to its appearance in a wide range of scientific and engineering areas. In this paper, we consider the class of synchronization problems over a closed subgroup of the orthogonal group, which covers many instances of group synchronization problems that arise in practice. Our contributions are threefold. First, we propose a unified approach to solve this class of group synchronization problems, which consists of a suitable initialization and an iterative refinement procedure via the generalized power method. Second, we derive a master theorem on the performance guarantee of the proposed approach. Under certain conditions on the subgroup, the measurement model, the noise model and the initialization, the estimation error of the iterates of our approach decreases geometrically. As our third contribution, we study concrete examples of the subgroup (including the orthogonal group, the special orthogonal group, the permutation group and the cyclic group), the measurement model, the noise model and the initialization. The validity of the related conditions in the master theorem are proved for these specific examples. Numerical experiments are also presented. Experiment results show that our approach outperforms existing approaches in terms of computational speed, scalability and estimation error