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Transfer entropy (TE) is a powerful tool for analyzing causality between time series and complex systems. However, it faces two key challenges. First, TE is often used to quantify the pairwise causal direction; yet, in real-world applications, one is always interested in identifying more complex causal relationships, such as indirect causation, common causation, and synergistic effect. Second, the estimation of TE usually relies on probability estimation, which is particularly complicated, or even infeasible for high-dimensional data. In this work, we take TE one step further and develop a pair of measures, the matrix-based conditional transfer entropy ( CTE M ) and the matrix-based high-order transfer entropy ( HTE M ). The former can detect both indirect and common causation, while the latter can detect synergistic effect. Making use of the recently proposed matrix-based Rényi's α -order entropy functional, CTE M and HTE M are defined on the eigenspectrum of a normalized Hermitian matrix of the projected data in kernel space, which avoids the necessity of density estimation and the curse of dimensionality. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of our measures in high-dimensional space, and their superiority in recovering complex causal structures for more than two time series. [ABSTRACT FROM AUTHOR]

Semi-supervised nonnegative matrix factorization (NMF) has received considerable attention in machine learning and data mining. A new semi-supervised NMF method, called dual semi-supervised convex nonnegative matrix factorization (DCNMF), is proposed in this paper for fully using the limited label information. Specifically, DCNMF simultaneously incorporates the pointwise and pairwise constraints of labeled samples as dual supervisory information into convex NMF, which results in a better low-dimensional data representation. Moreover, DCNMF imposes the nonnegative constraint only on the coefficient matrix but not on the base matrix. Consequently, DCNMF can process mixed-sign data, and hence enlarge the range of applications. We derive an efficient alternating iterative algorithm for DCNMF to solve the optimization, and analyze the proposed DCNMF method in terms of the convergence and computational complexity. We also discuss the relationships between DCNMF and several typical NMF based methods. Experimental results illustrate that DCNMF outperforms the related state-of-the-art NMF methods on nonnegative and mixed-sign datasets for clustering applications. [ABSTRACT FROM AUTHOR]