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Determining the number of clusters is one of the research questions attracting considerable interests in recent years. Majority of the existing methods require parametric assumptions and substantiated computations. In this paper we propose a simple yet powerful method for determining the number of clusters based on curvature. Our technique is computationally efficient and straightforward to implement. We compare our method with 6 other approaches on a wide range of simulated and real-world datasets. Theoretical motivation underlying the proposed method is also presented. [ABSTRACT FROM AUTHOR]

k nearest neighbors ( k NN) is one of the most widely used supervised learning algorithms to classify Gaussian distributed data, but it does not achieve good results when it is applied to nonlinear manifold distributed data, especially when a very limited amount of labeled samples are available. In this paper, we propose a new graph-based k NN algorithm which can effectively handle both Gaussian distributed data and nonlinear manifold distributed data. To achieve this goal, we first propose a constrained Tired Random Walk (TRW) by constructing an R -level nearest-neighbor strengthened tree over the graph, and then compute a TRW matrix for similarity measurement purposes. After this, the nearest neighbors are identified according to the TRW matrix and the class label of a query point is determined by the sum of all the TRW weights of its nearest neighbors. To deal with online situations, we also propose a new algorithm to handle sequential samples based a local neighborhood reconstruction. Comparison experiments are conducted on both synthetic data sets and real-world data sets to demonstrate the validity of the proposed new k NN algorithm and its improvements to other version of k NN algorithms. Given the widespread appearance of manifold structures in real-world problems and the popularity of the traditional k NN algorithm, the proposed manifold version k NN shows promising potential for classifying manifold-distributed data. [ABSTRACT FROM AUTHOR]