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Minimax Estimation of Distances on a Surface and Minimax Manifold Learning in the Isometric-to-Convex Setting

Published 25 Nov 2020 in stat.ML, cs.LG, math.ST, and stat.TH | (2011.12478v2)

Abstract: We start by considering the problem of estimating intrinsic distances on a smooth submanifold. We show that minimax optimality can be obtained via a reconstruction of the surface, and discuss the use of a particular mesh construction -- the tangential Delaunay complex -- for that purpose. We then turn to manifold learning and argue that a variant of Isomap where the distances are instead computed on a reconstructed surface is minimax optimal for the isometric variant of the problem.

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References (88)
  1. Optimal reach estimation and metric learning. arXiv preprint arXiv:2207.06074, 2022.
  2. E. Aamari and C. Levrard. Stability and minimax optimality of tangential Delaunay complexes for manifold reconstruction. Discrete & Computational Geometry, 59(4):923–971, 2018.
  3. C. Aaron and O. Bodart. Convergence rates for estimators of geodesic distances and fréchet expectations. Journal of Applied Probability, 55(4):1001–1013, 2018.
  4. N. M. Amato and G. Song. Using motion planning to study protein folding pathways. Journal of Computational Biology, 9(2):149–168, 2002.
  5. N. Amenta. The crust algorithm for 3-D surface reconstruction. In Symposium on Computational Geometry, pages 423–424, 1999.
  6. N. Amenta and M. Bern. Surface reconstruction by voronoi filtering. Discrete & Computational Geometry, 22(4):481–504, 1999.
  7. The crust and the beta-skeleton: Combinatorial curve reconstruction. In Graphical Models and Image Processing, pages 125–135, 1998.
  8. A new Voronoi-based surface reconstruction algorithm. In Proceedings of the 25th Annual Conference on Computer Graphics and Interactive Techniques, pages 415–421, 1998.
  9. A simple algorithm for homeomorphic surface reconstruction. In Proceedings of the Sixteenth Annual Symposium on Computational Geometry, pages 213–222, 2000.
  10. The power crust. In Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, pages 249–266, 2001.
  11. Perturbation bounds for procrustes, classical scaling, and trilateration, with applications to manifold learning. Journal of Machine Learning Research, 21:1–37, 2020.
  12. E. Arias-Castro and T. Le Gouic. Unconstrained and curvature-constrained shortest-path distances and their approximation. Discrete & Computational Geometry, 62(1):1–28, 2019.
  13. Spectral clustering based on local PCA. The Journal of Machine Learning Research, 18(1):253–309, 2017.
  14. E. Arias-Castro and B. Pelletier. On the convergence of maximum variance unfolding. The Journal of Machine Learning Research, 14(1):1747–1770, 2013.
  15. M. Balasubramanian and E. Schwartz. The isomap algorithm and topological stability. Science, 295(5552):7a, 2002.
  16. M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(16):1373–1396, 2003.
  17. M. Belkin and P. Niyogi. Towards a theoretical foundation for Laplacian-based manifold methods. Journal of Computer and System Sciences, 74(8):1289–1308, 2008.
  18. The ball-pivoting algorithm for surface reconstruction. IEEE Transactions on Visualization and Computer Graphics, 5(4):349–359, 1999.
  19. Graph approximations to geodesics on embedded manifolds. Technical report, Department of Psychology, Stanford University, 2000.
  20. J.-D. Boissonnat and F. Cazals. Smooth surface reconstruction via natural neighbour interpolation of distance functions. Computational Geometry, 22(1-3):185–203, 2002.
  21. Geometric and Topological Inference. Cambridge University Press, 2018.
  22. Delaunay triangulation of manifolds. Foundations of Computational Mathematics, 18(2):399–431, 2018.
  23. J.-D. Boissonnat and J. Flototto. A local coordinate system on a surface. In Proceedings of the Seventh ACM Symposium on Solid Modeling and Applications, SMA ’02, pages 116–126, New York, NY, USA, 2002. Association for Computing Machinery.
  24. J.-D. Boissonnat and J. Flötotto. A coordinate system associated with points scattered on a surface. Computer-Aided Design, 36(2):161–174, 2004.
  25. J.-D. Boissonnat and A. Ghosh. Manifold reconstruction using tangential Delaunay complexes. Discrete & Computational Geometry, 51(1):221–267, 2014.
  26. Isometric embeddings of the square flat torus in ambient space. Ensaios Matemáticos, 24:1–91, 2013.
  27. M. Brand. Charting a manifold. Advances in Neural Information Processing Systems, pages 985–992, 2003.
  28. J. Chen and Y. Han. Shortest paths on a polyhedron. In Proceedings of the Sixth Annual Symposium on Computational Geometry, pages 360–369, 1990.
  29. Sliver exudation. Journal of the ACM, 47(5):883–904, 2000.
  30. R. Coifman and S. Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, 2006.
  31. G. Csardi and T. Nepusz. The igraph software package for complex network research. Interjournal, Complex Systems:1695, 2006.
  32. V. de Silva and J. B. Tenenbaum. Sparse multidimensional scaling using landmark points. Technical report, Technical report, Stanford University, 2004.
  33. T. K. Dey and S. Goswami. Tight cocone: a water-tight surface reconstructor. Journal of Computing and Information Science in Engineering, 3(4):302–307, 2003.
  34. T. K. Dey and S. Goswami. Provable surface reconstruction from noisy samples. Computational Geometry, 35(1-2):124–141, 2006.
  35. Isotopic reconstruction of surfaces with boundaries. In Computer Graphics Forum, volume 28, pages 1371–1382. Wiley Online Library, 2009.
  36. J. Digne. An analysis and implementation of a parallel ball pivoting algorithm. Image Processing on Line, 4:149–168, 2014.
  37. V. Divol. Minimax adaptive estimation in manifold inference. Arxiv Preprint Arxiv:2001.04896, 2020.
  38. D. Donoho and C. Grimes. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proceedings of the National Academy of Sciences, 100(10):5591–5596, 2003.
  39. L. E. Dubins. On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents. American Journal of Mathematics, 79(3):497–516, 1957.
  40. Riemannian simplices and triangulations. Geometriae Dedicata, 179(1):91–138, 2015.
  41. H. Federer. Curvature measures. Transactions of the American Mathematical Society, 93(3):418–491, 1959.
  42. D. Freedman. Efficient simplicial reconstructions of manifolds from their samples. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(10):1349–1357, 2002.
  43. K. Fukunaga and D. R. Olsen. An algorithm for finding intrinsic dimensionality of data. IEEE Transactions on Computers, 100(2):176–183, 1971.
  44. Manifold estimation and singular deconvolution under Hausdorff loss. The Annals of Statistics, 40(2):941–963, 2012.
  45. E. Giné and V. Koltchinskii. Empirical graph Laplacian approximation of Laplace–Beltrami operators: Large sample results. In High Dimensional Probability, pages 238–259. Institute of Mathematical Statistics, 2006.
  46. Manifold learning: The price of normalization. Journal of Machine Learning Research, 9(Aug):1909–1939, 2008.
  47. J. C. Gower. Adding a point to vector diagrams in multivariate analysis. Biometrika, 55(3):582–585, 1968.
  48. From graphs to manifolds – weak and strong pointwise consistency of graph laplacians. In P. Auer and R. Meir, editors, Learning Theory, volume 3559 of Lecture Notes in Computer Science, pages 470–485. Springer Berlin / Heidelberg, 2005.
  49. Surface reconstruction from unorganized points. ACM SIGGRAPH Computer Graphics, 26(2):71–78, July 1992.
  50. C. Jamin. Tangential complex. In GUDHI User and Reference Manual. GUDHI Editorial Board, 3.2.0 edition, 2020.
  51. Deterministic sampling-based motion planning: Optimality, complexity, and performance. The International Journal of Robotics Research, 37(1):46–61, 2018.
  52. N. Kambhatla and T. K. Leen. Dimension reduction by local principal component analysis. Neural Computation, 9(7):1493–1516, 1997.
  53. L. Kaufman and P. Rousseeuw. Clustering by means of medoids. In Statistical Data Analysis Based on the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT Norm Conference, Neuchatel, 1987, pages 405–416, 1987.
  54. Triangulated surface mesh shortest paths. In CGAL User and Reference Manual. CGAL Editorial Board, 5.1 edition, 2020.
  55. Tight minimax rates for manifold estimation under Hausdorff loss. Electronic Journal of Statistics, 9(1):1562–1582, 2015.
  56. Minimax rates for estimating the dimension of a manifold. Journal of Computational Geometry, 10(1), 2019.
  57. Designing network diagrams. In Conference on Social Graphics, pages 22–50, 1980.
  58. J.-C. Latombe. Robot Motion Planning, volume 124. Springer, 2012.
  59. S. M. LaValle. Planning Algorithms. Cambridge University Press, 2006.
  60. D. Li and D. B. Dunson. Geodesic distance estimation with spherelets. Arxiv Preprint Arxiv:1907.00296, 2019.
  61. L. v. d. Maaten and G. Hinton. Visualizing data using t-SNE. Journal of Machine Learning Research, 9(Nov):2579–2605, 2008.
  62. Umap: Uniform manifold approximation and projection. Journal of Open Source Software, 3(29):861, 2018.
  63. Surface reconstruction for noisy point clouds. In Symposium on Geometry Processing, pages 53–62, 2005.
  64. D. Niculescu and B. Nath. DV based positioning in ad hoc networks. Telecommunication Systems, 22(1-4):267–280, 2003.
  65. Sensor network localization from local connectivity: Performance analysis for the MDS-map algorithm. In Information Theory, 2010 IEEE Information Theory Workshop On, pages 1–5. IEEE, 2010.
  66. A. Paprotny and J. Garcke. On a connection between maximum variance unfolding, shortest path problems and isomap. In Artificial Intelligence and Statistics, pages 859–867, 2012.
  67. A simple and fast algorithm for k-medoids clustering. Expert Systems with Applications, 36(2):3336–3341, 2009.
  68. S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500):2323–2326, 2000.
  69. J. W. Sammon. A nonlinear mapping for data structure analysis. IEEE Transactions on Computers, 100(5):401–409, 1969.
  70. A numerical solution to the generalized mapmaker’s problem: flattening nonconvex polyhedral surfaces. Pattern Analysis and Machine Intelligence, IEEE Transactions On, 11(9):1005–1008, 1989.
  71. Y. Shang and W. Ruml. Improved MDS-based localization. In Conference of the IEEE Computer and Communications Societies, volume 4, pages 2640–2651. IEEE, 2004.
  72. Localization from mere connectivity. In ACM International Symposium on Mobile Ad Hoc Networking and Computing, pages 201–212, 2003.
  73. V. Silva and J. Tenenbaum. Global versus local methods in nonlinear dimensionality reduction. Advances in Neural Information Processing Systems, 15:705–712, 2002.
  74. A. Singer. From graph to manifold Laplacian: The convergence rate. Applied and Computational Harmonic Analysis, 21(1):128–134, 2006.
  75. Convergence and rate of convergence of a manifold-based dimension reduction algorithm. In Advances in Neural Information Processing Systems, pages 1529–1536, 2008.
  76. Matrix Perturbation Theory. Computer Science and Scientific Computing. Academic Press Inc., Boston, MA, 1990.
  77. J. Tenenbaum. Mapping a manifold of perceptual observations. Advances in neural information processing systems, 10, 1997.
  78. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319–2323, 2000.
  79. Protein folding by motion planning. Physical Biology, 2(4):S148, 2005.
  80. W. S. Torgerson. Theory and Methods of Scaling. Wiley, 1958.
  81. Consistency of spectral clustering. The Annals of Statistics, 36(2):555–586, 2008.
  82. Learning a kernel matrix for nonlinear dimensionality reduction. In International Conference on Machine Learning, page 106, 2004.
  83. A. Weingessel and K. Hornik. Local pca algorithms. IEEE Transactions on Neural Networks, 11(6):1242–1250, 2000.
  84. H. Whitney. Geometric Integration Theory. Princeton University Press, 1957.
  85. Improving Chen and Han’s algorithm on the discrete geodesic problem. ACM Transactions on Graphics, 28(4):1–8, 2009.
  86. Q. Ye and W. Zhi. Discrete Hessian eigenmaps method for dimensionality reduction. Journal of Computational and Applied Mathematics, 278:197–212, 2015.
  87. H. Zha and Z. Zhang. Continuum isomap for manifold learnings. Computational Statistics & Data Analysis, 52(1):184–200, 2007.
  88. Z. Zhang and H. Zha. Principal manifolds and nonlinear dimension reduction via tangent space alignment. SIAM Journal on Scientific Computing, 26(1):313–338, 2004.
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