Riemannian Median and Its Estimation

In this paper, we define the geometric median of a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to calculate the median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also answers a recent question in P. T. Fletcher et al. [13

Suboptimal Choice Behaviour across Different Reinforcement Probabilities

Six adult roosters’ choice behaviour was investigated across a series of five experimental conditions and a series of replication of the same five experimental conditions. Stagner and Zentall (2010) found that pigeons prefer to choose an alternative with highly reliable discriminative stimuli but with less food reward over an alternative with non-discriminative stimuli but with more food reward. The current research systematically changed the probability of reinforcement associated with the discriminative stimulus through a series of experimental conditions. Experimental sessions were completed with six adult roosters. The experimental procedure was based on Stagner and Zentall’s (2010) experiment in which the suboptimal alternative with discriminative stimuli was associated with 100% reinforcement on 20% of the trials, and non-reinforcement on 80% of the trials; the optimal alternative with non-discriminative stimuli was associated with both 50% reinforcement on all trials. This research modified the probabilities of reinforcement associated with the discriminative alternative. In the first experimental condition, the probability of getting access to reinforcement was the same (50%) for each discriminative stimulus, thus, what was seen for the first time was that both alternatives were associated with non-discriminative stimuli. To insure reliability, a replication of the conditions was done after the first five experimental conditions were completed. The results showed that four of the roosters had suboptimal choice behaviour in the first five experimental conditions; however, only two of them maintained such suboptimal behaviour in the replication conditions. This result does not support the idea that the suboptimal choice behaviour with strong discriminative stimuli is a robust effect

Document Clustering Based On Max-Correntropy Non-Negative Matrix Factorization

Nonnegative matrix factorization (NMF) has been successfully applied to many areas for classification and clustering. Commonly-used NMF algorithms mainly target on minimizing the $l_2$ distance or Kullback-Leibler (KL) divergence, which may not be suitable for nonlinear case. In this paper, we propose a new decomposition method by maximizing the correntropy between the original and the product of two low-rank matrices for document clustering. This method also allows us to learn the new basis vectors of the semantic feature space from the data. To our knowledge, we haven't seen any work has been done by maximizing correntropy in NMF to cluster high dimensional document data. Our experiment results show the supremacy of our proposed method over other variants of NMF algorithm on Reuters21578 and TDT2 databasets.Comment: International Conference of Machine Learning and Cybernetics (ICMLC) 201

ASAP : towards accurate, stable and accelerative penetrating-rank estimation on large graphs

Pervasive web applications increasingly require a measure of similarity among objects. Penetrating-Rank (P-Rank) has been one of the promising link-based similarity metrics as it provides a comprehensive way of jointly encoding both incoming and outgoing links into computation for emerging applications. In this paper, we investigate P-Rank efficiency problem that encompasses its accuracy, stability and computational time. (1) We provide an accuracy estimate for iteratively computing P-Rank. A symmetric problem is to find the iteration number K needed for achieving a given accuracy ε. (2) We also analyze the stability of P-Rank, by showing that small choices of the damping factors would make P-Rank more stable and well-conditioned. (3) For undirected graphs, we also explicitly characterize the P-Rank solution in terms of matrices. This results in a novel non-iterative algorithm, termed ASAP , for efficiently computing P-Rank, which improves the CPU time from O(n 4) to O( n 3 ). Using real and synthetic data, we empirically verify the effectiveness and efficiency of our approaches