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Incremental Isometric Embedding of High-Dimensional Data Using Connected Neighborhood Graphs
January 2009 (vol. 31 no. 1)
pp. 86-98
Dongfang Zhao, Information, Distribution & Marketing, Inc., Atlanta
Li Yang, Western Michigan University, Kalamazoo
Most nonlinear data embedding methods use bottom-up approaches for capturing the underlying structure of data distributed on a manifold in high dimensional space. These methods often share the first step which defines neighbor points of every data point by building a connected neighborhood graph so that all data points can be embedded to a single coordinate system. These methods are required to work incrementally for dimensionality reduction in many applications. Because input data stream may be under-sampled or skewed from time to time, building connected neighborhood graph is crucial to the success of incremental data embedding using these methods. This paper presents algorithms for updating $k$-edge-connected and $k$-connected neighborhood graphs after a new data point is added or an old data point is deleted. It further utilizes a simple algorithm for updating all-pair shortest distances on the neighborhood graph. Together with incremental classical multidimensional scaling using iterative subspace approximation, this paper devises an incremental version of Isomap with enhancements to deal with under-sampled or unevenly distributed data. Experiments on both synthetic and real-world data sets show that the algorithm is efficient and maintains low dimensional configurations of high dimensional data under various data distributions.

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Index Terms:
Pattern Recognition, Models, Geometric, Statistical, Design Methodology, Feature evaluation and selection, Discrete Mathematics, Graph Theory, Graph algorithms, Database Management, Database Applications, Data mining.
Dongfang Zhao, Li Yang, "Incremental Isometric Embedding of High-Dimensional Data Using Connected Neighborhood Graphs," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 31, no. 1, pp. 86-98, Jan. 2009, doi:10.1109/TPAMI.2008.34
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