Xiao Wang

Hongwei Lin, Yunyang Xiong, Xiao Wang et al.

In this paper, we present the isogeometric least-squares collocation (IGA-L) method, which determines the numerical solution by making the approximate differential operator fit the real differential operator in a least-squares sense. The number of collocation points employed in IGA-L can be larger than that of the unknowns. Theoretical analysis and numerical examples presented in this paper show the superiority of IGA-L over state-of-the-art collocation methods. First, a small increase in the number of collocation points in IGA-L leads to a large improvement in the accuracy of its numerical solution. Second, IGA-L method is more flexible and more stable, because the number of collocation points in IGA-L is variable. Third, IGA-L is convergent in some cases of singular parameterization. Moreover, the consistency and convergence analysis are also developed in this paper.

Qiankun Shi, Xiao Wang, Hao Wang

In this paper, we study nonconvex constrained stochastic zeroth-order optimization problems, for which we have access to exact information of constraints and noisy function values of the objective. We propose a Bregman linearized augmented Lagrangian method that utilizes stochastic zeroth-order gradient estimators combined with a variance reduction technique. We analyze its oracle complexity, in terms of the total number of stochastic function value evaluations required to achieve an \(ε\)-KKT point in \(\ell_p\)-norm metrics with \(p \ge 2\), where \(p\) is a parameter associated with the selected Bregman distance. In particular, starting from a near-feasible initial point and using Rademacher smoothing, the oracle complexity is in order \(O(p d^{2/p} ε^{-3})\) for \(p \in [2, 2 \ln d]\), and \(O(\ln d \cdot ε^{-3})\) for \(p > 2 \ln d\), where \(d\) denotes the problem dimension. Those results show that the complexity of the proposed method can achieve a dimensional dependency lower than \(O(d)\) without requiring additional assumptions, provided that a Bregman distance is chosen properly. This offers a significant improvement in the high-dimensional setting over existing work, and matches the lowest complexity order with respect to the tolerance \(ε\) reported in the literature. Numerical experiments on constrained Lasso and black-box adversarial attack problems highlight the promising performances of the proposed method.

Simeng Qu, Xiao Wang

Dictionary learning is a cutting-edge area in imaging processing, that has recently led to state-of-the-art results in many signal processing tasks. The idea is to conduct a linear decomposition of a signal using a few atoms of a learned and usually over-completed dictionary instead of a pre-defined basis. Determining a proper size of the to-be-learned dictionary is crucial for both precision and efficiency of the process, while most of the existing dictionary learning algorithms choose the size quite arbitrarily. In this paper, a novel regularization method called the Grouped Smoothly Clipped Absolute Deviation (GSCAD) is employed for learning the dictionary. The proposed method can simultaneously learn a sparse dictionary and select the appropriate dictionary size. Efficient algorithm is designed based on the alternative direction method of multipliers (ADMM) which decomposes the joint non-convex problem with the non-convex penalty into two convex optimization problems. Several examples are presented for image denoising and the experimental results are compared with other state-of-the-art approaches.