Local Linear Regression on Manifolds and Its Geometric Interpretation

REFERENCES

• Allard , W. K. , Chen , G. and Maggioni , M. 2012 . Multi-Scale Geometric Methods for Data Sets II: Geometric Multi-Resolution Analysis . Applied and Computational Harmonic Analysis , 32 : 435 – 462 .
   Web of Science ®Google Scholar
• Aswani , A. , Bickel , P. and Tomlin , C. 2011 . Regression on Manifolds: Estimation of the Exterior Derivative . The Annals of Statistics , 39 : 48 – 81 .
   Web of Science ®Google Scholar
• Belkin , M. , Niyogi , P. and Sindhwani , V. 2006 . Manifold Regularization: A Geometric Framework for Learning From Labeled and Unlabeled Examples . The Journal of Machine Learning Research , 7 : 2399 – 2434 .
   Web of Science ®Google Scholar
• Bickel , P. J. and Li , B. 2007 . Local Polynomial Regression on Unknown Manifolds . Lecture Notes-Monograph Series , 54 : 177 – 186 .
   Google Scholar
• Carlsson , G. , Ishkhanov , T. , de Silva , V. and Zomorodian , A. 2008 . On the Local Behavior of Spaces of Natural Images . International Journal of Computer Vision , 76 : 1 – 12 .
   Web of Science ®Google Scholar
• Chen , L.-H. , Cheng , M.-Y. and Peng , L. 2009 . Conditional Variance Estimation in Heteroscedastic Regression Models . Journal of Statistical Planning and Inference , 139 : 236 – 245 .
   Web of Science ®Google Scholar
• Chikuse , Y. 2003 . Statistics on Special Manifolds , New York : Springer .
   Google Scholar
• Coifman , R. R. and Lafon , S. 2006 . Diffusion Maps . Applied and Computational Harmonic Analysis , 21 : 5 – 30 .
   Web of Science ®Google Scholar
• Fan , J. and Gijbels , I. 1996 . Local Polynomial Modelling and Its Applications , London: Chapman and Hall/CRC .
   Google Scholar
• Fan , J. and Li , R. 2001 . Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties . Journal of the American Statistical Association , 96 : 1348 – 1340 .
   Web of Science ®Google Scholar
• Fan , J. and Lv , J. 2008 . Sure Independence Screening for Ultrahigh Dimensional Feature Space . Journal of the Royal Statistical Society, Series B , 70 : 849 – 911 .
   PubMed Web of Science ®Google Scholar
• Fan , J. and Peng , H. 2004 . Nonconcave Penalized Likelihood With a Diverging Number of Parameters . The Annals of Statistics , 32 : 928 – 961 .
   Web of Science ®Google Scholar
• Fan , J. and Song , R. 2010 . Sure Independence Screening in Generalized Linear Models With np-Dimensionality . The Annals of Statistics , 38 : 3567 – 3604 .
   Web of Science ®Google Scholar
• Frank , A. and Asuncion , A. 2010 . UCI Machine Learning Repository . School of Information and Computer Science , available at http://archive.ics.uci.edu/mlIrvine, CA, University of California
   Google Scholar
• Frank , J. 2006 . Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in Their Native State (2nd ed.) , New York : Oxford University Press .
   Google Scholar
• Graf , F. , Kriegel , H.-P. , Schubert , M. , Pölsterl , S. and Cavallaro , A. 2011 . “ 2D Image Registration in CT Images Using Radial Image Descriptors ” . In MICCAI , 607 – 614 . Berlin Heidelberg : Springer-Verlag .
   Google Scholar
• Hall , P. , Marron , J. S. and Neeman , A. 2005 . Geometric Representation of High Dimension, Low Sample Size Data . Journal of the Royal Statistical Society, Series B , 67 : 427 – 444 .
   Google Scholar
• Kaslovsky , D. N. and Meyer , F. G. 2011 . Optimal Tangent Plane Recovery From Noisy Manifold Samples . CoRR, abs/1111.4601, available at http://dblp.uni-trier.de/db/journals/corr/corr1111.html#abs-1111-4601
   Google Scholar
• Lerman , G. and Zhang , T. 2011 . Robust Recovery of Multiple Subspaces by Geometric lp Minimization . The Annals of Statistics , 26 : 2686 – 2715 .
   Google Scholar
• Levina , E. and Bickel , P. J. 2005 . “ Maximum Likelihood Estimation of Intrinsic Dimension ” . In Advances in Neural Information Processing Systems (Vol. 17) , Edited by: Saul , L. , Weiss , Y. and Bottou , L. 777 – 784 . Cambridge, , MA : MIT Press .
   Google Scholar
• Li , R. and Liang , H. 2008 . Variable Selection in Semiparametric Regression Modeling . The Annals of Statistics , 36 : 261 – 286 .
   PubMed Web of Science ®Google Scholar
• Lin , B. , Zhang , C. and He , X. 2011 . Semi-Supervised Regression via Parallel Field Regularization . Neural Information Processing Systems , : 1 – 9 .
   Google Scholar
• Loubes , J.-M. and Pelletier , B. 2008 . A Kernel-Based Classifier on a Riemannian Manifold . Statutory Declaration , 26 : 35 – 51 .
   Google Scholar
• Mardia , K. and Jupp , P. 2000 . Directional Statistics , New York : Wiley .
   Google Scholar
• Mukherjee , S. , Wu , Q. and Zhou , D.-X. 2010 . Learning Gradients on Manifolds . Bernoulli , 16 : 181 – 207 .
   Web of Science ®Google Scholar
• Nadler , B. 2008 . Finite Sample Approximation Results for Principal Component Analysis: A Matrix Perturbation Approach . The Annals of Statistics , 36 : 2791 – 2817 .
   Web of Science ®Google Scholar
• Nilsson , J. , Sha , F. and Jordan , M. I. 2007 . Regression on Manifolds Using Kernel Dimension Reduction . Proceedings of the 24th International Conference on Machine Learning, ACM , : 697 – 704 .
   Google Scholar
• Niyogi , P. , Smale , S. and Weinberger , S. 2008 . Finding the Homology of Submanifolds With High Confidence From Random Samples . Discrete & Computational Geometry , 39 : 419 – 441 .
   Web of Science ®Google Scholar
• Pelletier , B. 2006 . Nonparametric Regression Estimation on Closed Riemannian Manifolds . Journal of Nonparametric Statistics , 18 : 57 – 67 .
   Web of Science ®Google Scholar
• Peyré , G. 2009 . Manifold Models for Signals and Images . Computer Vision and Image Understanding , 113 : 249 – 260 .
   Web of Science ®Google Scholar
• Ruppert , D. 1997 . Empirical-Bias Bandwidths for Local Polynomial Nonparametric Regression and Density Estimation . Journal of the American Statistical Association , 92 : 1049 – 1062 .
   Web of Science ®Google Scholar
• Ruppert , D. and Wand , M. P. 1994 . Multivariate Locally Weighted Least Squares Regression . The Annals of Statistics , 22 : 1346 – 1370 .
   Web of Science ®Google Scholar
• Singer , A. and Wu , H.-T. 2012 . Vector Diffusion Maps and the Connection Laplacian . Communications on Pure and Applied Mathematics , 65 : 1067 – 1144 .
   Web of Science ®Google Scholar
• Tenenbaum , J. B. , de Silva , V. and Langford , J. C. 2000 . A Global Geometric Framework for Nonlinear Dimensionality Reduction . Science , 290 : 2319 – 2323 .
   PubMed Web of Science ®Google Scholar
• Xia , Y. 2007 . A Constructive Approach to the Estimation of Dimension Reduction Directions . The Annals of Statistics , 35 : 2654 – 2690 .
   Web of Science ®Google Scholar
• Xia , Y. 2008 . A Multiple-index Model and Dimension Reduction . Journal of the American Statistical Association , 103 : 1631 – 1640 .
   Web of Science ®Google Scholar
• Zelnik-Manor , L. and Perona , P. 2004 . Self-Tuning Spectral Clustering . Advances in Neural Information Processing Systems , 2 : 1601 – 1608 .
   Google Scholar
• Zhang , C. , Jiang , Y. and Chai , Y. 2010 . Penalized Bregman Divergence for Large-Dimensional Regression and Classification . Biometrika , 97 : 551 – 560 .
   PubMed Web of Science ®Google Scholar
• Zhu , H. , Chen , Y. , Ibrahim , J. G. , Li , Y. and Lin , W. 2009 . Intrinsic Regression Models for Positive-Definite Matrices With Applications to Diffusion Tensor Imaging . Journal of the American Statistical Association , 104 : 1203 – 1212 .
   PubMed Web of Science ®Google Scholar