Presented is an algorithm for nonlinear dimensionality reduction that uses both local(short) and global(long) distances in order to learn the intrinsic geometry of manifolds with complicated topology. Since our algorithm matches nonlocal structures, it is robust even to strong noise. We show experimental results on both synthetic and real data demonstrating the advantages of our approach over stateof- the-art manifold learning methods.
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Belkin M. and Niyogi P. Laplacian eigenmaps and spectral techniques for em-bedding and clustering. In T. G. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14, pp. 585-591, MIT Press, Cambridge, MA, 2002.
Ben-Tal A. and Nemirovski A. S. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2001.
Bernstein M., de Silva V., Langford J. C., and Tenenbaum J. B. Graph ap-proximations to geodesics on embedded manifolds. Technical report, Stanford University, January 2001.
Blake A. and Zisserman A. Visual Reconstruction. The MIT Press, London, 1987.
Borg I. and Groenen P. Modern Multidimensional Scaling: Theory and Appli-cations. Springer, New York, 1997. xviii+471 pp.
Boult T. E. and Kender J. R. Visual surface reconstruction using sparse depth data. In Computer Vision and Pattern Recognition, volume 86, pp. 68-76, 1986.
Bronstein A. M., Bronstein M. M., and Kimmel R. Three-dimensional face recognition. International Journal of Computer Vision (IJCV), 64(1):5-30, August 2005.
Bronstein A. M., Bronstein M. M., and Kimmel R. Generalized multidimen-sional scaling: a framework for isometry-invariant partial surface matching. Pro-ceedings of the National Academy of Sciences, 103(5):1168-1172, January 2006.
Bronstein M. M., Bronstein A. M., and Kimmel R. Multigrid multidimensional scaling. Numerical Linear Algebra with Applications (NLAA), 13(2-3):149-171, March-April 2006.
Cabay S. and Jackson L. Polynomial extrapolation method for finding limits and antilimits of vector sequences. SIAM Journal on Numerical Analysis, 13 (5):734-752, October 1976.
Coifman R. R., Lafon S., Lee A. B., Maggioni M., Nadler B., Warner F., and Zucker S. W. Geometric diffusions as a tool for harmonic analysis and structure definition of data. Proceedings of the National Academy of Sciences, 102(21): 7426-7431, May 2005.
Cormen T. H., Leiserson C. E., and Rivest R. L. Introduction to Algorithms. MIT Press and McGraw-Hill, 1990.
Diaz R. M. and Arencibia A. Q., editors. Coloring of DT-MRI FIber Traces using Laplacian Eigenmaps, Las Palmas de Gran Canaria, Spain, Springer, February 24-28 2003.
Dijkstra E. W. A note on two problems in connection with graphs. Numerische Mathematik, 1:269-271, 1959.
Duda R. O., Hart P. E., and Stork D. G. Pattern Classification and Scene Analysis. Wiley-Interscience, 2nd edn, 2000.
Eddy R. P. Extrapolationg to the limit of a vector sequence. In P. C. Wang, edi-tor, Information Linkage Between Applied Mathematics and Industry, Academic Press, pp. 387-396. 1979.
Eldar Y., Lindenbaum M., Porat M., and Zeevi Y. The farthest point strategy for progressive image sampling. IEEE Transactions on Image Processing, 6(9): 1305-1315, September 1997.
Freedman D. Efficient simplicial reconstructions of manifolds from their sam-ples. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(10): 1349-1357, October 2002.
Grimes C. and Donoho D. L. When does isomap recover the natural parameter-ization of families of articulates images? Technical Report 2002-27, Department of Statistics, Stanford University, Stanford, CA 94305-4065, 2002.
Grimes C. and Donoho D. L. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proceedings of the National Academy of Sciences, 100(10):5591-5596, May 2003.
Guy G. and Medioni G. Inference of surfaces, 3D curves and junctions from sparse, noisy, 3D data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(11):1265-1277, November 1997.
Hu N., Huang W., and Ranganath S. Robust attentive behavior detection by non-linear head pose embedding and estimation. In A. Leonardis, H. Bischof and A. Pinz, editors, Proceedings of the 9th European Conference Computer Vision (ECCV), volume 3953, pp. 356-367, Graz, Austria, Springer. May 2006.
Kearsley A., Tapia R., and Trosset M. W. The solution of the metric stress and sstress problems in multidimensional scaling using newton’s method. Computa-tional Statistics, 13(3):369-396, 1998.
Keller Y., Lafon S., and Krauthammer M. Protein cluster analysis via directed diffusion. In The fifth Georgia Tech International Conference on Bioinformatics, November 2005.
Kimmel R. and Sethian J. A. Computing geodesic paths on manifolds. Proceed- ings of the National Academy of Sciences, 95(15):8431-8435, July 1998.
Lafon S. Diffusion Maps and Geometric Harmonics. Ph.D. dissertation, Grad-uate School of Yale University, May 2004.
Lafon S. and Lee A. B. Diffusion maps and coarse-graining: A unified framework for dimensionality reduction, graph partitioning and data set parameterization. IEEE transactions on Pattern Analysis and Machine Intelligence, 2006. To Appear.
Mordohai P. and Medioni G. Unsupervised dimensionality estimation and man- ifold learning in high-dimensional spaces by tensor voting. In Proceedings of International Joint Conference on Artificial Intelligence, pp. 798-803, 2005.
Pless R. Using Isomap to explore video sequences. In Proceedings of the 9th International Conference on Computer Vision, pp. 1433-1440, Nice, France, October 2003.
Roweis S. T. and Saul L. K. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323-2326, 2000.
Schwartz E. L., Shaw A., and Wolfson E. A numerical solution to the generalized mapmaker’s problem: Flattening nonconvex polyhedral surfaces. IEEE Trans-actions on Pattern Analysis and Machine Intelligence, 11:1005-1008, November 1989.
Sidi A. Efficient implementation of minimal polynomial and reduced rank ex-trapolation methods. Journal of Computational and Applied Mathematics, 36 (3):305-337, 1991.
Smith D. A., Ford W. F., and Sidi A. Extrapolation methods for vector se-quences. SIAM Review, 29(2):199-233, June 1987.
Tenenbaum J. B., de Silva V., and Langford J. C. A global geometric frame-work for nonlinear dimensionality reduction. Science, 290(5500):2319-2323, December 2000.
Trosset M. and Mathar R. On the existence of nonglobal minimizers of the stress criterion for metric multidimensional scaling. American Statistical Association: Proceedings Statistical Computing Section, pp. 158-162, November 1997.
Weinberger K. Q., Packer B. D., and Saul L. K. Nonlinear dimensionality reduc-tion by semidefinite programming and kernel matrix factorization. In Proceed-ings of the 10th International Workshop on Artificial Intelligence and Statistics, Barbados, January 2005.
WeinbergerK. Q. and SaulL. K. Unsupervised learning of image manifolds by semidefinite programming. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, vol. 2, pp. 988-995, Washington DC, IEEE Computer Society, 2004.
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Rosman, G., Bronstein, A.M., Bronstein, M.M., Kimmel, R. (2008). Topologically Constrained Isometric Embedding. In: Rosenhahn, B., Klette, R., Metaxas, D. (eds) Human Motion. Computational Imaging and Vision, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6693-1_10
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DOI: https://doi.org/10.1007/978-1-4020-6693-1_10
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