Chenglong BaoDepartment of Mathematics, National University of Singapore, Singapore,119076Jian-Feng CaiDepartment of Mathematics, University of Iowa, Iowa City, IA, USA, 52242Hui JiDepartment of Mathematics, National University of Singapore, Singapore,119076
In recent years, how to learn a dictionary from input images for sparse modelling has been one very active topic in image processing and recognition. Most existing dictionary learning methods consider an over-complete dictionary, e.g. the K-SVD method. Often they require solving some minimization problem that is very challenging in terms of computational feasibility and efficiency. However, if the correlations among dictionary atoms are not well constrained, the redundancy of the dictionary does not necessarily improve the performance of sparse coding. This paper proposed a fast orthogonal dictionary learning method for sparse image representation. With comparable performance on several image restoration tasks, the proposed method is much more computationally efficient than the over-complete dictionary based learning methods.
Chenglong BaoDepartment of Mathematics, National University of Singapore, Singapore, 119076Hui JiDepartment of Mathematics, National University of Singapore, Singapore, 119076Yuhui QuanDepartment of Mathematics, National University of Singapore, Singapore, 119076Zuowei ShenDepartment of Mathematics, National University of Singapore, Singapore, 119076
Sparse coding and dictionary learning have seen their applications in many vision tasks, which usually is formulated as a non-convex optimization problem. Many iterative methods have been proposed to tackle such an optimization problem. However, it remains an open problem to have a method that is not only practically fast but also is globally convergent. In this paper, we proposed a fast proximal method for solving \ell_0 norm based dictionary learning problems, and we proved that the whole sequence generated by the proposed method converges to a stationary point with sub-linear convergence rate. The benefit of having a fast
and convergent dictionary learning method is demonstrated in the applications of image recovery and face recognition.
Chenglong BaoDepartment of Mathematics, National University of Singapore, SingaporeYuhui QuanDepartment of Mathematics, National University of Singapore, SingaporeHui JiDepartment of Mathematics, National University of Singapore, Singapore
Recently, sparse coding has been widely used in many applications ranging from image recovery to pattern recognition. The low mutual coherence of a dictionary is an important property that ensures the optimality of the sparse code generated from this dictionary. Indeed, most existing dictionary learning methods for sparse coding either implicitly or explicitly tried to learn an incoherent dictionary, which requires solving a very challenging non-convex optimization problem. In this paper, we proposed a hybrid alternating proximal algorithm for incoherent dictionary learning, and established its global convergence property. Such a convergent incoherent dictionary learning method is not only of theoretical interest, but also might benefit many sparse coding based applications.
Yuhui QuanSchool of Computer Science & Engineering, South China Univ. of Tech., Guangzhou 510006, China; Department of Mathematics, National University of Singapore, Singapore 117542Chenglong BaoDepartment of Mathematics, National University of Singapore, Singapore 117542Hui JiDepartment of Mathematics, National University of Singapore, Singapore 117542
Most existing dictionary learning algorithms consider a linear sparse model, which often cannot effectively characterize the nonlinear properties present in many types of visual data, e.g. dynamic texture (DT). Such nonlinear properties can be exploited by the so-called kernel sparse coding. This paper proposed an equiangular kernel dictionary learning method with optimal mutual coherence to exploit the nonlinear sparsity of high-dimensional visual data. Two main issues are addressed in the proposed method: (1) coding stability for redundant dictionary of infinite-dimensional space; and (2) computational efficiency for computing kernel matrix of training samples of high-dimensional data. The proposed kernel sparse coding method is applied to dynamic texture analysis with both local DT pattern extraction and global DT pattern characterization. The experimental results showed its performance gain over existing methods.
Zihao WangBNRist, Department of Computer Science and Technology, RIIT, Institute of Internet Industry, Tsinghua UniversityDatong ZhouDepartment of Mathematical Sciences, Tsinghua UniversityMing YangDepartment of Computer Science and Technology, Tsinghua UniversityYong ZhangBNRist, Department of Computer Science and Technology, RIIT, Institute of Internet Industry, Tsinghua UniversityChenglong BaoYau Mathematical Sciences Center, Tsinghua UniversityHao WuDepartment of Mathematical Sciences, Tsinghua University
Computing the distance among linguistic objects is an essential problem in natural language processing. The word mover’s distance (WMD) has been successfully applied to measure the document distance by synthesizing the low-level word similarity with the framework of optimal transport (OT). However, due to the global transportation nature of OT, the WMD may overestimate the semantic dissimilarity when documents contain unequal semantic details. In this paper, we propose to address this overestimation issue with a novel Wasserstein-Fisher-Rao (WFR) document distance grounded on unbalanced optimal transport theory. Compared to the WMD, the WFR document distance provides a trade-off between global transportation and local truncation, which leads to a better similarity measure for unequal semantic details. Moreover, an efficient prune strategy is particularly designed for the WFR document distance to facilitate the top-k queries among a large number of documents. Extensive experimental results show that the WFR document distance achieves higher accuracy that WMD and even its supervised variation s-WMD.