Topologically Constrained Isometric Embedding

— We present a new algorithm for nonlinear dimensionality reduction that consistently uses global information, which enables understanding the intrinsic geometry of non-convex manifolds. Compared to methods that consider only local information, our method appears to be more robust to noise. We demonstrate the performance of our algorithm and compare it to state-of-the-art methods on synthetic as well as real data.

[1]  Alexander M. Bronstein,et al.  Multigrid multidimensional scaling , 2006, Numer. Linear Algebra Appl..

[2]  S. Cabay,et al.  A Polynomial Extrapolation Method for Finding Limits and Antilimits of Vector Sequences , 1976 .

[3]  Surendra Ranganath,et al.  Robust Attentive Behavior Detection by Non-linear Head Pose Embedding and Estimation , 2006, ECCV.

[4]  J. Tenenbaum,et al.  A global geometric framework for nonlinear dimensionality reduction. , 2000, Science.

[5]  Eric L. Schwartz,et al.  A Numerical Solution to the Generalized Mapmaker's Problem: Flattening Nonconvex Polyhedral Surfaces , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[6]  Anthony J. Kearsley,et al.  The Solution of the Metric STRESS and SSTRESS Problems in Multidimensional Scaling Using Newton's Method , 1995 .

[7]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[8]  David L. Donoho,et al.  Image Manifolds which are Isometric to Euclidean Space , 2005, Journal of Mathematical Imaging and Vision.

[9]  Ron Kimmel,et al.  Representation Analysis and Synthesis of Lip Images Using Dimensionality Reduction , 2006, International Journal of Computer Vision.

[10]  Carl-Fredrik Westin,et al.  Coloring of DT-MRI Fiber Traces Using Laplacian Eigenmaps , 2003, EUROCAST.

[11]  Peter E. Hart,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[12]  Ann B. Lee,et al.  Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[13]  Kilian Q. Weinberger,et al.  Unsupervised Learning of Image Manifolds by Semidefinite Programming , 2004, CVPR.

[14]  Daniel Freedman,et al.  Efficient Simplicial Reconstructions of Manifolds from Their Samples , 2002, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Gérard G. Medioni,et al.  Unsupervised Dimensionality Estimation and Manifold Learning in high-dimensional Spaces by Tensor Voting , 2005, IJCAI.

[16]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[17]  D. Donoho,et al.  Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[18]  I. Hassan.  Embedded , 2005, The Cyber Security Handbook.

[19]  Kilian Q. Weinberger,et al.  Unsupervised learning of image manifolds by semidefinite programming , 2004, CVPR 2004.

[20]  S T Roweis,et al.  Nonlinear dimensionality reduction by locally linear embedding. , 2000, Science.

[21]  Alexander M. Bronstein,et al.  Three-Dimensional Face Recognition , 2005, International Journal of Computer Vision.

[22]  R. P. Eddy.  EXTRAPOLATING TO THE LIMIT OF A VECTOR SEQUENCE , 1979 .

[23]  Mikhail Belkin,et al.  Laplacian Eigenmaps and Spectral Techniques for Embedding and Clustering , 2001, NIPS.

[24]  Yehoshua Y. Zeevi,et al.  The farthest point strategy for progressive image sampling , 1997, IEEE Trans. Image Process..

[25]  Avram Sidi,et al.  Efficient implementation of minimal polynomial and reduced rank extrapolation methods , 1991 .

[26]  Ron Kimmel,et al.  Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[27]  Gérard G. Medioni,et al.  Inference of Surfaces, 3D Curves, and Junctions From Sparse, Noisy, 3D Data , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[28]  Mikhail Belkin,et al.  Laplacian Eigenmaps for Dimensionality Reduction and Data Representation , 2003, Neural Computation.

[29]  John R. Kender,et al.  Visual Surface Reconstruction Using Sparse Depth Data , 1986, CVPR 1986.

[30]  Robert Pless,et al.  Image spaces and video trajectories: using Isomap to explore video sequences , 2003, Proceedings Ninth IEEE International Conference on Computer Vision.

[31]  A. Sidi,et al.  Extrapolation methods for vector sequences , 1987 .

[32]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[33]  Ann B. Lee,et al.  Diffusion maps and coarse-graining: a unified framework for dimensionality reduction, graph partitioning, and data set parameterization , 2006, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[34]  Andrew Blake,et al.  Visual Reconstruction , 1987, Deep Learning for EEG-Based Brain–Computer Interfaces.

[35]  Richard O. Duda,et al.  Pattern classification and scene analysis , 1974, A Wiley-Interscience publication.

[36]  Kilian Q. Weinberger,et al.  Nonlinear Dimensionality Reduction by Semidefinite Programming and Kernel Matrix Factorization , 2005, AISTATS.

[37]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[38]  P. Groenen,et al.  Modern Multidimensional Scaling: Theory and Applications , 1999 .