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Abstract: In this paper, we propose a general learning framework based on local and global regularization. In the local regularization part, our algorithm constructs a regularized classifier for each data point using its neighborhood, while the global regularization part adopts a Laplacian regularizer to smooth the data labels predicted by those local classifiers. We show that such a learning framework can easily be incorporated into either unsupervised learning, semi-supervised learning, and supervised learning paradigm. Moreover, many existing learning algorithms can be derived from our framework. Finally we present some experimental results to show the effectiveness of our method. [Copyright &y& Elsevier]

Abstract: Measuring the structural similarity of graphs is a challenging and outstanding problem. Most of the classical approaches of the so-called exact graph matching methods are based on graph or subgraph isomorphic relations of the underlying graphs. In contrast to these methods in this paper we introduce a novel approach to measure the structural similarity of directed and undirected graphs that is mainly based on margins of feature vectors representing graphs. We introduce novel graph similarity and dissimilarity measures, provide some properties and analyze their algorithmic complexity. We find that the computational complexity of our measures is polynomial in the graph size and, hence, significantly better than classical methods from, e.g. exact graph matching which are NP-complete. Numerically, we provide some examples of our measure and compare the results with the well-known graph edit distance. [Copyright &y& Elsevier]

Abstract: In this paper, we use resource-bounded dimension theory to investigate polynomial size circuits. We show that for every , has ith-order scaled -strong dimension 0. We also show that has -dimension and -strong dimension 1. Our results improve previous measure results of Lutz [Almost everywhere high nonuniform complexity, J. Comput. Syst. Sci. 44(2) (1992) 220–258] and dimension results of Hitchcock and Vinodchandran [Dimension, entropy rates, and compression, in: Proc. 19th IEEE Conf. Computational Complexity, 2004, pp. 174–183, J. Comput. Syst. Sci., to appear]. Additionally, we establish a Supergale Dilation Theorem, which extends the martingale dilation technique introduced implicitly by Ambos-Spies, Terwijn, and Zheng [Resource bounded randomness and weakly complete problems, Theoret. Comput. Sci. 172(1–2) (1997) 195–207] and made explicit by Juedes and Lutz [Weak completeness in E and , Theoret. Comput. Sci. 143(1) (1995) 149–158]. [Copyright &y& Elsevier]