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Distance-Preserving Projection of High-Dimensional Data for Nonlinear Dimensionality Reduction
September 2004 (vol. 26 no. 9)
pp. 1243-1246
Li Yang, IEEE
A distance-preserving method is presented to map high-dimensional data sequentially to low-dimensional space. It preserves exact distances of each data point to its nearest neighbor and to some other near neighbors. Intrinsic dimensionality of data is estimated by examining the preservation of interpoint distances. The method has no user-selectable parameter. It can successfully project data when the data points are spread among multiple clusters. Results of experiments show its usefulness in projecting high-dimensional data.

[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. 7, Jan. 2002.
[2] R.S. Bennet, The Intrinsic Dimensionality of Signal Collections IEEE Trans. Information Theory, vol. 15, no. 5, pp. 517-525, Sept. 1969.
[3] G. Biswas, A.K. Jain, and R.C. Dubes, Evaluation of Projection Algorithms IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 3, no. 6, pp. 701-708, Nov. 1981.
[4] C.L. Blake and C.J. Merz, UCI Repository of Machine Learning Databases http://www.ics.uci.edu/~mlearnMLRepository.html , 1998.
[5] J. Bruske and G. Sommer, “Intrinsic Dimensionality Estimation with Optimally Topology Preserving Maps,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 5, pp. 572-575, May 1998.
[6] 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, pp. 1,197-1,206, 1997.
[7] K. Fukunaga and D.R. Olsen, An Algorithm for Finding Intrinsic Dimensionality of Data IEEE Trans. Computers, vol. 20, no. 2, pp. 176-183, Feb. 1971.
[8] J.J.W. Sammon, A Nonlinear Mapping for Data Structure Analysis IEEE Trans. Computers, vol. 18, no. 5, pp. 401-409, May 1969.
[9] A.K. Jain and R.C. Dubes, Algorithms for Clustering Data. Prentice-Hall, 1988.
[10] A.K. Jain, R.P.W. Duin, and J. Mao, Statistical Pattern Recognition: A Review IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 22, no. 1, pp. 4-37, Jan. 2000.
[11] T. Kohonen, Self-Organizing Maps, second edition. Springer, 1997.
[12] J. Kruskal, Multidimensional Scaling by Optimizing Goodness-of-Fit to a Nonmetric Hypothesis Psychometrika, vol. 29, no. 1, pp. 1-27, Mar. 1964.
[13] J. Kruskal, Comments on a Nonlinear Mapping for Data Structure Alanysis IEEE Trans. Computers, vol. 20, no. 12, p. 1614, Dec. 1971.
[14] Y. LeCun, MNIST Database http://yann.lecun.com/exdbmnist/, 1998.
[15] J.A. Lee, A. Lendasse, N. Donckers, and M. Verleysen, A Robust Nonlinear Projection Method Proc. Eighth European Symp. Artificial Neural Networks (ESANN 2000), pp. 13-20, Apr. 2000.
[16] R.C.T. Lee, J.R. Slagle, and H. Blum, A Triangulation Method for the Sequential Mapping of Points from N-Space to Two-Space IEEE Trans. Computers, vol. 26, no. 3, pp. 288-292, Mar. 1977.
[17] H. Niemann and J. Weiss, A Fast Converging Algorithm for Nonlinear Mapping of High-Dimensional Data onto a Plane IEEE Trans. Computers, vol. 28, no. 2, pp. 142-147, Feb. 1979.
[18] D.R. Olsen and K. Fukunaga, Representation of Nonlinear Data Surfaces IEEE Trans. Computers, vol. 22, no. 10, pp. 915-922, Oct. 1973.
[19] K.W. Pettis, T.A. Bailey, A.K. Jain, and R.C. Dubes, An Intrinsic Dimensionality Estimator from Near-Neighbor Information IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 1, no. 1, pp. 25-37, Jan. 1979.
[20] R.C. Prim, Shortest Connection Networks and Some Generalizations Bell System Technical J., vol. 36, pp. 1389-1401, Nov. 1957.
[21] S.T. Roweis and L.K. Saul, Nonlinear Dimensionality Reduction by Locally Linear Embedding Science, vol. 290, pp. 2323-2326, Dec. 2000.
[22] H.S. Seung and D. Lee, The Manifold Ways of Perception Science, vol. 290, pp. 2268-2269, Dec. 2000.
[23] 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.
[24] G.V. Trunk, Statistical Estimation of the Intrinsic Dimensionality of a Noisy Signal Collection IEEE Trans. Computers, vol. 25, no. 2, pp. 165-171, 1976.
[25] P.J. Verveer and R.P.W. Duin, “An Evaluation of Intrinsic Dimensionality Estimators,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 17, no. 1, pp. 81-86, Jan. 1995.
[26] M. Vlachos, C. Domeniconi, D. Gunopulos, G. Kollios, and N. Koudas, Non-Linear Dimensionality Reduction Techniques for Classification and Visualization Proc. Eighth ACM KDD Conf. Knowledge Discovery and Data Mining, pp. 645-651, July 2002.

Index Terms:
Pattern recognition, statistical, feature evaluation and selection, pattern analysis.
Citation:
Li Yang, "Distance-Preserving Projection of High-Dimensional Data for Nonlinear Dimensionality Reduction," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 26, no. 9, pp. 1243-1246, Sept. 2004, doi:10.1109/TPAMI.2004.66
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