Nonlinear Dimensionality Reduction for Data with Disconnected Neighborhood Graph

Abstract

Neighborhood graph based nonlinear dimensionality reduction algorithms, such as Isomap and LLE, perform well under an assumption that the neighborhood graph is connected. However, for datasets consisting of multiple clusters or lying on multiple manifolds, the neighborhood graphs are often disconnected, or in other words, have multiple connected components. Neighborhood graph based dimensionality reduction techniques cannot recognize both the local and global properties of such datasets. In this paper, a new method, called enhanced neighborhood graph, is proposed to solve the problem. The concept is to add edges to the neighborhood graph adaptively and iteratively until it becomes connected. Nonlinear dimensionality reduction can then be performed based on the enhanced neighborhood graph. As a result, both local and global properties of the data can be exactly recognized. In this study, thorough simulations on synthetic datasets and natural datasets are conducted. The experimental results corroborate that the proposed method provides significant improvements on dimensionality reduction for data with disconnected neighborhood graph.

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Acknowledgements

This work is partially supported by the National Science Foundation of China under Grant no.61601112, the Fundamental Research Funds for the Central Universities and DHU Distinguished Young Professor Program. This work is also partially supported by the Natural Science Foundation of China under grant no. 61572156.

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Correspondence to Tommy W. S. Chow.

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Fan, J., Chow, T.W.S., Zhao, M. et al. Nonlinear Dimensionality Reduction for Data with Disconnected Neighborhood Graph. Neural Process Lett 47, 697–716 (2018). https://doi.org/10.1007/s11063-017-9676-5

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Keywords

  • Nonlinear dimensionality reduction
  • Isomap
  • LLE
  • Disconnected graph
  • Enhanced neighborhood graph
  • Multiple manifolds