This Article 
 Bibliographic References 
 Add to: 
Building k-Connected Neighborhood Graphs for Isometric Data Embedding
May 2006 (vol. 28 no. 5)
pp. 827-831
Isometric data embedding using geodesic distance requires the construction of a connected neighborhood graph so that the geodesic distance between every pair of data points can be estimated. This paper proposes an approach for constructing k-connected neighborhood graphs. The approach works by applying a greedy algorithm to add each edge, in a nondecreasing order of edge length, to a neighborhood graph if end vertices of the edge are not yet k-connected on the graph. The k-connectedness between vertices is tested using a network flow technique by assigning every vertex a unit flow capacity. This approach is applicable to a wide range of data. Experiments show that it gives better estimation of geodesic distances than other approaches, especially when the data are under-sampled or nonuniformly distributed.

[1] M. Balasubramanian, E.L. Schwartz, J.B. Tenenbaum, V. de Silva, and J.C. Langford, “The Isomap Algorithm and Topological Stability: Technical Comments,” Science, vol. 295, no. 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] E.A. Dinic, “Algorithm for Solution of a Problem of Maximum Flow in a Network with Power Estimation,” Soviet Math. Doklady, vol. 11, pp. 1277-1280, 1970.
[5] J. Edmonds and R.M. Karp, “Theoretical Improvements in Algorithmic Efficiency for Network Flow Problems,” J. ACM, vol. 19, no. 2, pp. 248-264, Apr. 1972.
[6] S. Even and R.E. Tarjan, “Network Flow and Testing Graph Connectivity,” SIAM J. Computing, vol. 4, no. 4, pp. 507-518, Dec. 1975.
[7] L.R. Ford,Jr. and D.R. Fulkerson, Flows in Networks. Princeton Univ. Press, 1962.
[8] 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.
[9] J.J.W. Sammon, “A Nonlinear Mapping for Data Structure Analysis,” IEEE Trans. Computers, vol. 18, no. 5, pp. 401-409, May 1969.
[10] J. Kruskal, “Multidimensional Scaling by Optimizing Goodness-of-Fit to a Nonmetric Hypothesis,” Psychometrika, vol. 29, pp. 1-27, 1964.
[11] J. Kruskal, “Comments on a Monlinear Mapping for Data Structure Analysis,” IEEE Trans. Computers, vol. 20, no. 12, p. 1614, Dec. 1971.
[12] J.A. Lee, A. Lendasse, and M. Verleysen, “Nonlinear Projection with Curvilinear Distances: Isomap versus Curvilinear Distance Anaylsis,” Neurocomputing, vol. 57, pp. 49-76, Mar. 2004.
[13] D. Matula, “k-Blocks and Ultrablocks in Graphs,” J. Combinatorial Theory, Series B, vol. 24, no. 1, pp. 1-13, Feb. 1978.
[14] R.E. Tarjan, “Testing Graph Connectivity,” Proc. Sixth Ann. ACM Symp. Theory of Computing, pp. 185-193, 1974.
[15] 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.
[16] L. Yang, “Distance-Preserving Projection of High Dimensional Data for Nonlinear Dimensionality Reduction,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 26, no. 9, pp. 1243-1246, Sept. 2004.
[17] 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.
[18] L. Yang, “ Building k-Edge-Connected Neighborhood Graphs for Distance-Based Data Projection,” Pattern Recognition Letters, vol. 26, no. 13, pp. 2015-2021, Oct. 2005.
[19] L. Yang, “Building k Edge-Disjoint Spanning Trees of Minimum Total Length for Isometric Data Embedding,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 27, no. 10, pp. 1680-1683, Oct. 2005.
[20] H. Zha and Z. Zhang, “Isometric Embedding and Continuum Isomap,” Proc. 20th Int'l Conf. Machine Learning, pp. 864-871, Aug. 2003.

Index Terms:
Data embedding, graph connectivity, manifold learning, network flow.
Li Yang, "Building k-Connected Neighborhood Graphs for Isometric Data Embedding," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 28, no. 5, pp. 827-831, May 2006, doi:10.1109/TPAMI.2006.89
Usage of this product signifies your acceptance of the Terms of Use.