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Building k Edge-Disjoint Spanning Trees of Minimum Total Length for Isometric Data Embedding
October 2005 (vol. 27 no. 10)
pp. 1680-1683
Li Yang, IEEE
Isometric data embedding requires construction of a neighborhood graph that spans all data points so that geodesic distance between any pair of data points could be estimated by distance along the shortest path between the pair on the graph. This paper presents an approach for constructing k-edge-connected neighborhood graphs. It works by finding k edge-disjoint spanning trees the sum of whose total lengths is a minimum. Experiments show that it outperforms the nearest neighbor approach for geodesic distance estimation.

[1] M. Balasubramanian, E.L. Schwartz, J.B. Tenenbaum, V. de Silva, and J.C. Langford, “The Isomap Algorithm and Topological Stability,” Science, vol. 295, p. 7a, Jan. 2002.
[2] T.F. Cox and M.A.A. Cox, Multidimensional Scaling, second ed. Chapman & Hall, 2001.
[3] P. Demartines and J. Herault, “Curvilinear Component Analysis: A Self-Organizing Neural Network for Nonlinear Mapping of Data Sets,” IEEE Trans. Neural Networks, vol. 8, no. 1, pp. 148-154, Jan. 1997.
[4] M.R. Garay and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W.H. Freeman and Company, 1979.
[5] J.J.W. Sammon, “A Nonlinear Mapping for Data Structure Analysis,” IEEE Trans. Computers, vol. 18, no. 5, pp. 401-409, May 1969.
[6] J. Kruskal, “On the Shortest Spanning Subtree of a Graph and the Travelling Salesman Problem,” Proc. Am. Math. Soc., vol. 7, no. 1, pp. 48-50, 1956.
[7] J.A. Lee, A. Lendasse, N. Donckers, and M. Verleysen, “A Robust Nonlinear Projection Method,” Proc. Eighth European Symp. Artificial Neural Networks, pp. 13-20, Apr. 2000.
[8] S.T. Roweis and L.K. Saul, “Nonlinear Dimensionality Reduction by Locally Linear Embedding,” Science, vol. 290, pp. 2323-2326, Dec. 2000.
[9] H.S. Seung and D. Lee, “The Manifold Ways of Perception,” Science, vol. 290, pp. 2268-2269, Dec. 2000.
[10] J.B. Tenenbaum, V. de Silva, and J.C. Langford, “A Global Geometric Framework for Nonlinear Dimensionality Reduction,” Science, vol. 290, pp. 2319-2323, Dec. 2000.
[11] L. Yang, “$k$ -Edge Connected Neighborhood Graph for Geodesic Distance Estimation and Nonlinear Data Projection,” Proc. 17th Int'l Conf. Pattern Recognition, vol. 1, pp. 196-199, Aug. 2004.
[12] H. Zha and Z. Zhang, “Isometric Embedding and Continuum ISOMAP,” Proc. 20th Int'l Conf. Machine Learning, pp. 864-871, Aug. 2003.

Index Terms:
Index Terms- Data embedding, dimensionality reduction, manifold learning, minimum spanning tree, neighborhood graph.
Li Yang, "Building k Edge-Disjoint Spanning Trees of Minimum Total Length for Isometric Data Embedding," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 10, pp. 1680-1683, Oct. 2005, doi:10.1109/TPAMI.2005.192
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