In this paper, we propose an interpretation of active learning from a pure algebraic view and combine it with semi-supervised manifold learning. The proposed active manifold learning algorithm aims to learn the low-dimensional parameter space of the manifold with high accuracy from smartly labeled samples. We demonstrate that this problem is equivalent to a condition number minimization problem of the alignment matrix. Focusing on this problem, we first give a theoretical upper bound for the solution. Then we develop a heuristic but effective sample selection algorithm with the help of the Gershgorin circle theorem. We investigate the rationality, the feasibility, the universality and the complexity of the proposed method and demonstrate that our method yields encouraging active learning results.

In this paper, we propose a novel dictionary learning method in the semi-supervised setting by dynamically coupling graph and group structures. To this end, samples are represented by sparse codes inheriting their graph structure while the labeled samples within the same class are represented with group sparsity, sharing the same atoms of the dictionary. Instead of statically combining graph and group structures, we take advantage of them in a mutually reinforcing way — in the dictionary learning phase, we introduce the unlabeled samples into groups by an entropy-based method and then update the corresponding local graph, resulting in a more structured and discriminative dictionary. We analyze the relationship between the two structures and prove the convergence of our proposed method. Focusing on image classification task, we evaluate our approach on several datasets and obtain superior performance compared with the state-of-the-art methods, especially in the case of only a few labeled samples and limited dictionary size.