Skip to main content
Log in

The Geometry of Synchronization Problems and Learning Group Actions

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

We develop a geometric framework, based on the classical theory of fibre bundles, to characterize the cohomological nature of a large class of synchronization-type problems in the context of graph inference and combinatorial optimization. We identify each synchronization problem in topological group G on connected graph \(\Gamma \) with a flat principal G-bundle over \(\Gamma \), thus establishing a classification result for synchronization problems using the representation variety of the fundamental group of \(\Gamma \) into G. We then develop a twisted Hodge theory on flat vector bundles associated with these flat principal G-bundles, and provide a geometric realization of the graph connection Laplacian as the lowest-degree Hodge Laplacian in the twisted de Rham–Hodge cochain complex. Motivated by these geometric intuitions, we propose to study the problem of learning group actions—partitioning a collection of objects based on the local synchronizability of pairwise correspondence relations—and provide a heuristic synchronization-based algorithm for solving this type of problems. We demonstrate the efficacy of this algorithm on simulated and real datasets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Notes

  1. Recall (see, e.g. [145, Sect. 2]) that a fibre bundle \(\pi :{\mathfrak {B}}\rightarrow X\), with total space \({\mathfrak {B}}\) and base space X, is said to be flat if it admits a system of local trivializations with locally constant bundle coordinate transformations.

  2. Note that a flat bundle is not necessarily trivial (i.e. isomorphic to a product space)—the fundamental group of the base space plays a central role in this development (see e.g. [115, Chap. 2]).

  3. Note this is not the case for jointly analyzing a collection of shapes in a landmark-based Procrustes analysis framework; see e.g. [34].

References

  1. Aigerman, N., Poranne, R., Lipman, Y.: Seamless surface mappings. ACM Trans. Graph. (TOG) 34(4), 72 (2015)

    Google Scholar 

  2. Al-Aifari, R., Daubechies, I., Lipman, Y.: Continuous procrustes distance between two surfaces. Commun. Pure Appl. Math. 66(6), 934–964 (2013). https://doi.org/10.1002/cpa.21444

    Article  MathSciNet  MATH  Google Scholar 

  3. Alon, N., Karp, R.M., Peleg, D., West, D.: A graph-theoretic game and its application to the k-server problem. SIAM J. Comput. 24(1), 78–100 (1995)

    MathSciNet  MATH  Google Scholar 

  4. Angenent, S., Haker, S., Tannenbaum, A., Kikinis, R.: On the Laplace-Beltrami operator and brain surface flattening. IEEE Trans. Med. Imaging 18(8), 700–711 (1999). https://doi.org/10.1109/42.796283

    Article  Google Scholar 

  5. Anosov, D.V., Bolibruch, A.A.: The Riemann–Hilbert problem. Aspects of Mathematics, vol. 22. Vieweg, Braunschweig (1994)

  6. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Atiyah, M.F., Bott, R.: The Yang–Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 308(1505), 523–615 (1983)

  8. Aubry, M., Schlickewei, U., Cremers, D.: The wave kernel signature: a quantum mechanical approach to shape analysis. In: 2011 IEEE International Conference on Computer Vision Workshops (ICCV Workshops), pp. 1626–1633 (2011). https://doi.org/10.1109/ICCVW.2011.6130444

  9. Bajaj, C., Gao, T., He, Z., Huang, Q., Liang, Z.: SMAC: simultaneous mapping and clustering using spectral decompositions. In: Proceedings of the 35th International Conference on Machine Learning, vol. 80, pp. 324–333 (2018). http://proceedings.mlr.press/v80/bajaj18a.html

  10. Bandeira, A.S., Charikar, M., Singer, A., Zhu, A.: Multireference alignment using semidefinite programming. In: Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, pp. 459–470. ACM, New York (2014)

  11. Bandeira, A.S., Chen, Y., Singer, A.: Non-unique Games over Compact Groups and Orientation Estimation in Cryo-EM (2015). arXiv:1505.03840

  12. Bandeira, A.S., Kennedy, C., Singer, A.: Approximating the Little Grothendieck Problem over the Orthogonal and Unitary Groups. Mathematical Programming, pp. 1–43 (2016). https://doi.org/10.1007/s10107-016-0993-7

  13. Bandeira, A.S., Singer, A., Spielman, D.A.: A cheeger inequality for the graph connection Laplacian. SIAM J. Matrix Anal. Appl. 34(4), 1611–1630 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Bandelt, H.J., Chepoi, V.: Metric graph theory and geometry: a survey. Contemp. Math. 453, 49–86 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003). https://doi.org/10.1162/089976603321780317

    Article  MATH  Google Scholar 

  16. Belkin, M., Niyogi, P.: Semi-supervised learning on Riemannian manifolds. Mach. Learn. 56(1–3), 209–239 (2004)

    MATH  Google Scholar 

  17. Blitzstein, J., Diaconis, P.: A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math. 6(4), 489–522 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Bobenko, A.I., Sullivan, J.M., Schröder, P., Ziegler, G.: Discrete Differential Geometry. Springer, Heidelberg (2008)

    MATH  Google Scholar 

  19. Bolibrukh, A.A.: The Riemann-Hilbert problem. Russ. Math. Surv. 45(2), 11–58 (1990)

    MathSciNet  MATH  Google Scholar 

  20. Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol. 82. Springer, New York (1982)

  21. Boumal, N., Singer, A., Absil, P.A., Blondel, V.D.: Cramér-Rao bounds for synchronization of rotations. Inf. Inference 3(1), 1–39 (2014)

    MathSciNet  MATH  Google Scholar 

  22. Boyer, D.M.: Relief index of second mandibular molars is a correlate of diet among prosimian primates and other euarchontan mammals. J. Hum. Evol. 55(6), 1118–1137 (2008)

    Google Scholar 

  23. Boyer, D.M., Lipman, Y., St. Clair, E., Puente, J., Patel, B.A., Funkhouser, T., Jernvall, J., Daubechies, I.: Algorithms to automatically quantify the geometric similarity of anatomical surfaces. Proc. Natl. Acad. Sci. U.S.A. 108(45), 18,221–18,226 (2011). https://doi.org/10.1073/pnas.1112822108

  24. Boyer, D.M., Puente, J., Gladman, J.T., Glynn, C., Mukherjee, S., Yapuncich, G.S., Daubechies, I.: A new fully automated approach for aligning and comparing shapes. Anat. Rec. 298(1), 249–276 (2015)

    Google Scholar 

  25. Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups. Graduate Texts in Mathematics. Springer, Heidelberg (2003)

  26. Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Numerical Geometry of Non-Rigid Shapes. Springer, New York (2008)

    MATH  Google Scholar 

  27. Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1704–1711. IEEE (2010)

  28. Brown, R.: Groupoids and van Kampen’s Theorem. Proc. Lond. Math. Soc. (3) 17, pp. 385–401. Citeseer (1967)

  29. Brown, R., Higgins, P., Sivera, R.: Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids. EMS Series of Lectures in Mathematics. European Mathematical Society, Zurich (2011)

  30. Brylinski, J.L.: Loop Spaces, Characteristic Classes and Geometric Quantization, vol. 107. Springer, New York (2007)

    MATH  Google Scholar 

  31. Bunn, J.M., Boyer, D.M., Lipman, Y., St Clair, E.M., Jernvall, J., Daubechies, I.: Comparing Dirichlet normal surface energy of tooth crowns, a new technique of molar shape quantification for dietary inference, with previous methods in isolation and in combination. Am. J. Phys. Anthropol. 145(2), 247–261 (2011)

    Google Scholar 

  32. Candogan, O., Menache, I., Ozdaglar, A., Parrilo, P.A.: Flows and decompositions of games: harmonic and potential games. Math. Oper. Res. 36(3), 474–503 (2011). https://doi.org/10.1287/moor.1110.0500

    Article  MathSciNet  MATH  Google Scholar 

  33. Carlsson, G., Zomorodian, A., Collins, A., Guibas, L.J.: Persistence barcodes for shapes. Int. J. Shape Model. 11(02), 149–187 (2005)

    MATH  Google Scholar 

  34. Chaudhury, K.N., Khoo, Y., Singer, A.: Global registration of multiple point clouds using semidefinite programming. SIAM J. Optim. 25(1), 468–501 (2015)

    MathSciNet  MATH  Google Scholar 

  35. Cheeger, J.: A lower bound for the smallest eigenvalue of the Laplacian. In: Problems in Analysis (Papers dedicated to Salomon Bochner, 1969), pp. 195–199. Princeton University Press, Princeton (1970)

  36. Chen, Y., Guibas, L., Huang, Q.: Near-optimal joint object matching via convex relaxation. In: Jebara, T., Xing, E.P. (eds.) Proceedings of the 31st International Conference on Machine Learning (ICML-14) (JMLR Workshop and Conference Proceedings ), pp. 100–108 (2014)

  37. Chentsov, N.N.: A systematic theory of exponential families of probability distributions. Theor. Probab. Appl. 11, 425–425 (1966)

    MathSciNet  MATH  Google Scholar 

  38. Chung, F.R.: Spectral Graph Theory No. 92. CBMS Regional Conference Series in Mathematics. American Mathematical Society, Providence (1997)

  39. Chung, F.R.: Four proofs for the Cheeger inequality and graph partition algorithms. Proc. ICCM 2, 378 (2007)

    Google Scholar 

  40. Chung, F.R., Lu, L., Vu, V.: Spectra of random graphs with given expected degrees. Proc. Natl. Acad. Sci. U.S.A. 100(11), 6313–6318 (2003). https://doi.org/10.1073/pnas.0937490100

    Article  MathSciNet  MATH  Google Scholar 

  41. Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006). https://doi.org/10.1016/j.acha.2006.04.006. Special Issue: Diffusion Maps and Wavelets

  42. Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Nadler, B., Warner, F., Zucker, S.W.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc. Natl. Acad. Sci. U.S.A. 102(21), 7426–7431 (2005). https://doi.org/10.1073/pnas.0500334102

    Article  MATH  Google Scholar 

  43. Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Nadler, B., Warner, F., Zucker, S.W.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: multiscale methods. Proc. Natl. Acad. Sci. U.S.A. 102(21), 7432–7437 (2005). https://doi.org/10.1073/pnas.0500896102

    Article  MATH  Google Scholar 

  44. Coja-Oghlan, A., Lanka, A.: The spectral gap of random graphs with given expected degrees. International Colloquium on Automata. Languages, and Programming, pp. 15–26. Springer, Heidelberg (2006)

  45. Connes, A.: Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math. 62(1), 41–144 (1985). https://doi.org/10.1007/BF02698807

    Article  MathSciNet  MATH  Google Scholar 

  46. Connes, A.: Noncommutative geometry. In: Alon, N., et al. (eds.) Visions in Mathematics, pp. 481–559. Birkhauser, Basel (2000)

    Google Scholar 

  47. Corlette, K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28, 361–382 (1988)

    MathSciNet  MATH  Google Scholar 

  48. Cramér, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946)

    MATH  Google Scholar 

  49. Crane, K., de Goes, F., Desbrun, M., Schröder, P.: Digital geometry processing with discrete exterior calculus. In: ACM SIGGRAPH 2013 Courses, SIGGRAPH’13, pp. 7:1–7:126. ACM, New York (2013). https://doi.org/10.1145/2504435.2504442

  50. Cucuringu, M.: Sync-Rank: robust ranking, constrained ranking and rank aggregation via eigenvector and SDP synchronization. IEEE Trans. Netw. Sci. Eng. 3(1), 58–79 (2016)

    MathSciNet  Google Scholar 

  51. Cucuringu, M., Lipman, Y., Singer, A.: Sensor network localization by eigenvector synchronization over the Euclidean group. ACM Trans. Sensor Netw. (TOSN) 8(3), 19 (2012)

    Google Scholar 

  52. Deligne, P.: Équations Différentielles à Points Singuliers Réguliers. Lecture Notes in Mathematics, vol. 163. Springer, Berlin (1970)

  53. Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete Exterior Calculus (2005). arXiv:math/0508341

  54. Donoho, D.L., Grimes, C.: Hessian eigenmaps: new locally linear embedding techniques for high-dimensional data. Proc. Natl. Acad. Sci. U.S.A. 100, 5591–5596 (2003)

    MATH  Google Scholar 

  55. Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis, vol. 4. Wiley, New York (1998)

    MATH  Google Scholar 

  56. Dupont, J.L.: Simplicial de Rham cohomology and characteristic classes of flat bundles. Topology 15(3), 233–245 (1976)

    MathSciNet  MATH  Google Scholar 

  57. Edelsbrunner, H., Harer, J.: Computational Topology: An Introduction. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  58. El Karoui, N., Wu, H.-T.: Graph connection Laplacian methods can be made robust to noise. Ann. Stat. 44(1), 346–372 (2016). https://doi.org/10.1214/14-AOS1275

    Article  MathSciNet  MATH  Google Scholar 

  59. Esnault, H.: Characteristic classes of flat bundles. Topology 27(3), 323–352 (1988)

    MathSciNet  MATH  Google Scholar 

  60. Evans, A.R., Wilson, G.P., Fortelius, M., Jernvall, J.: High-level similarity of dentitions in carnivorans and rodents. Nature 445(7123), 78–81 (2007)

    Google Scholar 

  61. Fanuel, M., Alaíz, C.M., Suykens, J.A.K.: Magnetic Eigenmaps for Community Detection in Directed Networks (2016). arXiv:1606.07359

  62. Fanuel, M., Suykens, J.A.K.: Deformed Laplacians and spectral ranking in directed networks (2015). arXiv:1511.00492

  63. Félix, Y., Lavendhomme, R.: On de Rham’s theorem in synthetic differential geometry. J. Pure and Appl. Algebra 69(1), 21–31 (1990)

    MathSciNet  MATH  Google Scholar 

  64. Fisher, R.A.: On the mathematical foundations of theoretical statistics. Philos. Trans. R. Soc. Lond. A 222, 309–368 (1922)

    MATH  Google Scholar 

  65. Gao, T.: Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric Morphometrics. Ph.D. Thesis, Duke University (2015)

  66. Gao, T.: The diffusion geometry of fibre bundles: horizontal diffusion maps. submitted (2016) arXiv:1602.02330

  67. Gao, T., Kovalsky, S.Z., Boyer, D.M., Daubechies, I.: Gaussian process landmarking for three-dimensional geometric morphometrics. SIAM J. Math. Data Sci. 1(1), 237–267 (2019). https://doi.org/10.1137/18M1203481

    Article  MathSciNet  Google Scholar 

  68. Gao, T., Kovalsky, S.Z., Daubechies, I.: Gaussian process landmarking on manifolds. SIAM J. Math. Data Sci. 1(1), 208–236 (2019). https://doi.org/10.1137/18M1184035

    Article  MathSciNet  Google Scholar 

  69. Gao, T., Yapuncich, G.S., Daubechies, I., Mukherjee, S., Boyer, D.M.: Development and assessment of fully automated and globally transitive geometric morphometric methods, with application to a biological comparative dataset with high interspecific variation. Anat. Rec. 301(4), 636–658 (2018). https://doi.org/10.1002/ar.23700

    Article  Google Scholar 

  70. García-Raboso, A., Rayan, S.: Introduction to nonabelian hodge theory. Calabi-Yau Varieties: Arithmetic. Geometry and Physics, pp. 131–171. Springer, Cham (2015)

  71. Goldman, W.M.: Characteristic classes and representations of discrete subgroups of lie groups. Bull. Am. Math. Soc. 6(1), 91–94 (1982)

    MathSciNet  MATH  Google Scholar 

  72. Goldman, W.M.: Mapping class group dynamics on surface group representations. Problems on Mapping Class Groups and Related Topics. Proceedings of the Symposium on Pure Mathematics, vol. 74, pp. 189–214. American Mathematical Society, Providence (2006)

  73. Gonzalez, P.N., Barbeito-Andrés, J., D’Addona, L.A., Bernal, V., Perez, S.I.: Technical Note: performance of semi- and fully automated approaches for registration of 3D surface coordinates in geometric morphometric studies. Am. J. Phys. Anthropol. 160(1), 169–178 (2016)

    Google Scholar 

  74. Gower, J.C.: Generalized procrustes analysis. Psychometrika 40(1), 33–51 (1975). https://doi.org/10.1007/BF02291478

    Article  MathSciNet  MATH  Google Scholar 

  75. Gower, J.C., Dijksterhuis, G.B.: Procrustes Problems. Oxford Statistical Science Series, vol. 3. Oxford University Press, Oxford (2004)

  76. Haefliger, A.: Complexes of Groups and Orbihedra. Group Theory From a Geometrical Viewpoint. In: Proceedings of a Workshop, Held at the International Centre for Theoretical Physics in Trieste, Italy, 26 March-6 April 1990, pp. 504–540. World Scientific, Singapore (1991)

  77. Haefliger, A.: Extension of complexes of groups. Ann. Inst. Fourier Grenoblé 42(1–2), 275–311 (1992)

    MathSciNet  MATH  Google Scholar 

  78. Hartley, R., Trumpf, J., Dai, Y., Li, H.: Rotation averaging. Int. J. Comput. Vis. 103(3), 267–305 (2013). https://doi.org/10.1007/s11263-012-0601-0

    Article  MathSciNet  MATH  Google Scholar 

  79. Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55(3), 59–126 (1987)

    MathSciNet  MATH  Google Scholar 

  80. Hitchin, N.J.: Flat connections and geometric quantization. Commun. Math. Phys. 131(2), 347–380 (1990)

    MathSciNet  MATH  Google Scholar 

  81. Hoffman, C., Kahle, M., Paquette, E.: Spectral Gaps of Random Graphs and Applications to Random Topology. arXiv preprint arXiv:1201.0425 (2012)

  82. Huang, Q., Guibas, L.: Consistent shape maps via semidefinite programming. computer graphics Forum. In: Proceedings of Eurographics Symposium on Geometry Processing (SGP) 32(5), 177–186 (2013)

  83. Huang, Q., Wang, F., Guibas, L.: Functional map networks for analyzing and exploring large shape collections. ACM Trans. Graph. (TOG) 33(4), 36 (2014)

    MATH  Google Scholar 

  84. Huang, Q., Zhang, G., Gao, L., Hu, S., Bustcher, A., Guibas, L.: An optimization approach for extracting and encoding consistent maps in a shape collection. ACM Trans. Graph. 31, 125:1–125:11 (2012)

  85. Jiang, X., Lim, L.H., Yao, Y., Ye, Y.: Statistical ranking and combinatorial Hodge theory. Math. Progr. 127(1), 203–244 (2011)

    MathSciNet  MATH  Google Scholar 

  86. Johnson, J.L., Goldring, T.: Discrete Hodge theory on graphs: a tutorial. Comput. Sci. Eng. 15(5), 42–55 (2013)

    Google Scholar 

  87. Kamber, F.W., Tondeur, P.: Flat bundles and characteristic classes of group-representations. Am. J. Math. 89(4), 857–886 (1967)

    MathSciNet  MATH  Google Scholar 

  88. Kashiwara, M.: Faisceaux Constructibles et Systèmes Holonomes d’équations aux Dérivées Partielles Linéaires à Points Singuliers Réguliers. Séminaire Équations aux dérivées partielles (Polytechnique) pp. 1–6 (1979)

  89. Kashiwara, M.: The Riemann-Hilbert problem for holonomic systems. Publ. Res. Inst. Math. Sci. 20(2), 319–365 (1984)

    MathSciNet  MATH  Google Scholar 

  90. Kenyon, R.: Spanning forests and the vector bundle Laplacian. Ann. Probab. 39(5), 1983–2017 (2011)

    MathSciNet  MATH  Google Scholar 

  91. Kezurer, I., Kovalsky, S.Z., Basri, R., Lipman, Y.: Tight relaxation of quadratic matching. Comput. Graph. Forum 34(5), 115–128 (2015). https://doi.org/10.1111/cgf.12701

    Article  Google Scholar 

  92. Kock, A.: Differential forms with values in groups. Bull. Aust. Math. Soc. 25(03), 357–386 (1982)

    MathSciNet  MATH  Google Scholar 

  93. Kock, A.: Synthetic Differential Geometry, vol. 333. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  94. Koehl, P., Hass, J.: Landmark-free geometric methods in biological shape analysis. J. R. Soc. Interface 12(113), 20150,795 (2015)

  95. Kokkinos, I., Bronstein, M.M., Litman, R., Bronstein, A.M.: Intrinsic shape context descriptors for deformable shapes. In: 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 159–166. IEEE (2012)

  96. Kovalsky, S.Z., Aigerman, N., Basri, R., Lipman, Y.: Controlling singular values with semidefinite programming. ACM Trans. Graph. 33(4), 69–71 (2014)

    MATH  Google Scholar 

  97. Kyng, R., Lee, Y.T., Peng, R., Sachdeva, S., Spielman, D.A.: Sparsified Cholesky and multigrid solvers for connection Laplacians. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pp. 842–850. ACM, New York (2016). https://doi.org/10.1145/2897518.2897640

  98. Labourie, F.: Lectures on Representations of Surface Groups. Zurich Lectures in Advanced Mathematics. European Mathematical Society (2013)

  99. Lai, R., Zhao, H.: Multi-Scale Non-Rigid Point Cloud Registration Using Robust Sliced-Wasserstein Distance via Laplace–Beltrami Eigenmap. arXiv preprint arXiv:1406.3758 (2014)

  100. Lee, J.M.: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol. 218. Springer, New York (2003)

  101. Lim, L.H.: Hodge Laplacians on Graphs. arXiv preprint arXiv:1507.05379 (2015)

  102. Lipman, Y., Daubechies, I.: Conformal Wasserstein distances: comparing surfaces in polynomial time. Adv. Math. 227(3), 1047–1077 (2011). https://doi.org/10.1016/j.aim.2011.01.020

    Article  MathSciNet  MATH  Google Scholar 

  103. Lipman, Y., Puente, J., Daubechies, I.: Conformal Wasserstein distance: II. Computational aspects and extensions. Math. Comput. 82(281), 331–381 (2013)

    MATH  Google Scholar 

  104. Lue, H.S.: Characteristic classes for the deformation of flat connections. Trans. Am. Math. Soc. 217, 379–393 (1976)

    MathSciNet  MATH  Google Scholar 

  105. Madore, J.: An Introduction to Noncommutative Differential Geometry and Its Physical Applications, vol. 257. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  106. Majid, S.: Noncommutative Riemannian geometry on graphs. J. Geom. Phys. 69, 74–93 (2013)

    MathSciNet  MATH  Google Scholar 

  107. Malhanobis, P.C.: On the generalized distance in statistics. Proc. Natl. Inst. Sci. India 2, 49–55 (1936)

    Google Scholar 

  108. Maron, H., Dym, N., Kezurer, I., Kovalsky, S., Lipman, Y.: Point registration via efficient convex relaxation. ACM Trans. Graph. (TOG) 35(4), 73 (2016)

    Google Scholar 

  109. Martinec, D., Pajdla, T.: Robust rotation and translation estimation in multiview reconstruction. In: 2007 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1–8. IEEE (2007)

  110. Mebkhout, Z.: Sur le Probleme de Hilbert-Riemann. Complex Analysis. Microlocal Calculus and Relativistic Quantum Theory, pp. 90–110. Springer, Berlin (1980)

  111. Mebkhout, Z.: Une Autre Équivalence de catégories. Compos. Math. 51(1), 63–88 (1984)

    MATH  Google Scholar 

  112. Michor, P.W.: Topics in Differential Geometry, vol. 93. American Mathematical Society, Providence (2008)

    MATH  Google Scholar 

  113. Milnor, J.: On the existence of a connection with curvature zero. Comment. Math. Helv. 32(1), 215–223 (1958)

    MathSciNet  MATH  Google Scholar 

  114. Milnor, J., Stasheff, J.D.: Characteristic classes. In: Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton (1974)

  115. Morita, S.: Geometry of Characteristic Classes, vol. 199. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  116. Mukherjee, S., Steenbergen, J.: Random walks on simplicial complexes and harmonics. Random Struct. Algorithms 49(2), 379–405 (2016)

    MathSciNet  MATH  Google Scholar 

  117. Naor, A., Regev, O., Vidick, T.: Efficient rounding for the noncommutative Grothendieck inequality. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of computing, pp. 71–80. ACM, New York (2013)

  118. Nguyen, A., Ben-Chen, M., Welnicka, K., Ye, Y., Guibas, L.: An optimization approach to improving collections of shape maps. Comput. Graph. Forum 30(5), 1481–1491 (2011)

    Google Scholar 

  119. Pampush, J.D., Winchester, J.M., Morse, P.E., Vining, A.Q., Boyer, D.M., Kay, R.F.: Introducing molaR: a new R package for quantitative topographic analysis of teeth (and other topographic surfaces). J. Mamm. Evol. 23, 397–412 (2016)

    Google Scholar 

  120. Parzanchevski, O., Rosenthal, R.: Simplicial complexes: spectrum, homology and random walks. Random Struct. Algorithms 50(2), 225–261 (2016)

    MathSciNet  MATH  Google Scholar 

  121. Parzanchevski, O., Rosenthal, R., Tessler, R.J.: Isoperimetric inequalities in simplicial complexes. Combinatorica 36(2), 195–227 (2015)

    MathSciNet  MATH  Google Scholar 

  122. Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)

    MathSciNet  MATH  Google Scholar 

  123. Puente, J.: Distances and Algorithms to Compare Sets of Shapes for Automated Biological Morphometrics. Ph.D. thesis, Princeton University Press, Princeton (2013)

  124. Rangarajan, A., Chui, H., Bookstein, F.L.: The softassign procrustes matching algorithm. In: Information Processing in Medical Imaging, pp. 29–42. Springer, New York (1997)

  125. Rao, C.R.: Information and accuracy obtainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91 (1945)

    MathSciNet  Google Scholar 

  126. Raviv, D., Bronstein, M.M., Bronstein, A.M., Kimmel, R.: Volumetric heat kernel signatures. In: Proceedings of the ACM Workshop on 3D Object Retrieval, pp. 39–44. ACM, New York (2010)

  127. Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)

    Google Scholar 

  128. Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)

    Google Scholar 

  129. Shkolnisky, Y., Singer, A.: Viewing direction estimation in Cryo-EM using synchronization. SIAM J. Imaging Sci. 5(3), 1088–1110 (2012)

    MathSciNet  MATH  Google Scholar 

  130. Simpson, C.T.: Higgs bundles and local systems. Publ. Math. l’IHÉS 75(1), 5–95 (1992)

    MathSciNet  MATH  Google Scholar 

  131. Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math. l’IHÉS 79(1), 47–129 (1994)

    MathSciNet  MATH  Google Scholar 

  132. Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety. II. Publ. Math. Inst. Hautes Sci. 80(1), 5–79 (1994). https://doi.org/10.1007/BF02698895

    Article  MathSciNet  MATH  Google Scholar 

  133. Singer, A.: Angular synchronization by eigenvectors and semidefinite programming. Appl. Comput. Harm. Anal. 30(1), 20–36 (2011). https://doi.org/10.1016/j.acha.2010.02.001

    Article  MathSciNet  MATH  Google Scholar 

  134. Singer, A.: Ht, Wu: Orientability and diffusion maps. Appl. Computat. Harmon. Anal. 31(1), 44–58 (2011). https://doi.org/10.1016/j.acha.2010.10.001

    Article  MATH  Google Scholar 

  135. Singer, A.: Ht, Wu: Vector diffusion maps and the connection Laplacian. Commun. Pure Appl. Math. 65(8), 1067–1144 (2012). https://doi.org/10.1002/cpa.21395

    Article  MATH  Google Scholar 

  136. Singer, A., Wu, H.-T.: Spectral Convergence of the Connection Laplacian from Random Samples. arXiv preprint arXiv:1306.1587 (2013)

  137. Singer, A., Zhao, Z., Shkolnisky, Y., Hadani, R.: Viewing angle classification of cryo-electron microscopy images using eigenvectors. SIAM J. Imaging Sci. 4(2), 723–759 (2011)

    MathSciNet  MATH  Google Scholar 

  138. Solomon, J., Nguyen, A., Butscher, A., Ben-Chen, M., Guibas, L.: Soft maps between surfaces. Comput. Graph. Forum 31(5), 1617–1626 (2012). https://doi.org/10.1111/j.1467-8659.2012.03167.x

    Article  Google Scholar 

  139. Steenbergen, J., Klivans, C., Mukherjee, S.: A Cheeger-type inequality on simplicial complexes. Adv. Appl. Math. 56, 56–77 (2014)

    MathSciNet  MATH  Google Scholar 

  140. Steenrod, N.E.: The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton (1951)

  141. Su, Z., Zeng, W., Shi, R., Wang, Y., Sun, J., Gu, X.: Area preserving brain mapping. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2235–2242 (2013)

  142. Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. Comput. Graph. Forum 28(5), 1383–1392 (2009)

    Google Scholar 

  143. Taubes, C.: Differential Geometry: Bundles, Connections, Metrics and Curvature, vol. 23. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  144. Tenenbaum, J.B., de Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)

    Google Scholar 

  145. Tu, L.W.: Hodge Theory and the Local Torelli Problem. Memoirs of the American Mathematical Society, vol. 43(279). American Mathematical Society, Providence (1983)

  146. Tzveneva, T., Singer, A., Rusinkiewicz, S.: Global Alignment of Multiple 3-D Scans using Eigenvector Synchronization (bachelor thesis). Technical Report, Princeton University Press, Princeton, Tech. Rep. (2011)

    Google Scholar 

  147. van der Maaten, L., Hinton, G.: Visualizing data using t-SNE. J. Mach. Learn. Res. 9(Nov), 2579–2605 (2008)

  148. Vassiliou, E.: Flat bundles and holonomy homomorphisms. Manuscr. Math. 42(2–3), 161–170 (1983)

    MathSciNet  MATH  Google Scholar 

  149. Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics. American Mathematical Society, Providence (2003)

    MATH  Google Scholar 

  150. Villani, C.: Optimal Transport: Old and New, 2009th edn. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2008)

    MATH  Google Scholar 

  151. Vitek, N.S., Manz, C.L., Gao, T., Bloch, J.I., Strait, S.G., Boyer, D.M.: Semi-supervised determination of pseudocryptic morphotypes using observer-free characterizations of anatomical alignment and shape. Ecol. Evol. 7(14), 5041–5055 (2017)

    Google Scholar 

  152. Von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)

    MathSciNet  Google Scholar 

  153. Wang, F., Huang, Q., Guibas, L.: Image co-segmentation via consistent functional maps. In: 2013 IEEE International Conference on Computer Vision (ICCV), pp. 849–856. IEEE (2013)

  154. Wang, L., Singer, A.: Exact and stable recovery of rotations for robust synchronization. Inf. Inference 2(2), 145–193 (2013). https://doi.org/10.1093/imaiai/iat005

    Article  MathSciNet  MATH  Google Scholar 

  155. Weinstein, A.: The symplectic structure on moduli space. In: The Floer Memorial Volume, pp. 627–635. Birkhäuser, Basel (1995)

  156. Wells, R.O.: Differential Analysis on Complex Manifolds. Graduate Texts in Mathematics, vol. 65. Springer, Berlin (2007)

  157. Xia, E.Z.: Abelian and non-abelian cohomology. Geometry, Topology and Dynamics of Character Varieties 23, 309–349 (2012)

    MathSciNet  MATH  Google Scholar 

  158. Ye, K., Lim, L.H.: Cohomology of cryo-electron microscopy. SIAM J. Appl. Algebra Geom. 1(1), 507–535 (2017). https://doi.org/10.1137/16M1070220

    Article  MathSciNet  MATH  Google Scholar 

  159. Zhao, X., Su, Z., Gu, X., Kaufman, A., Sun, J., Gao, J., Luo, F.: Area-preservation mapping using optimal mass transport. IEEE Trans. Vis. Comput. Graph. 19(12), 2838–2847 (2013). https://doi.org/10.1109/TVCG.2013.135

    Article  Google Scholar 

  160. Zhu, L., Haker, S., Tannenbaum, A.: Area-preserving mappings for the visualization of medical structures. In: Ellis, R., Peters, T. (eds.) Medical Image Computing and Computer-Assisted Intervention (MICCAI 2003). Lecture Notes in Computer Science, vol. 2879, pp. 277–284. Springer, Berlin (2003). https://doi.org/10.1007/978-3-540-39903-2_35

Download references

Acknowledgements

The authors would like to thank Pankaj Agarwal, Douglas Boyer, Robert Bryant, Ingrid Daubechies, Pawe l D lotko, Kathryn Hess, Lek-Heng Lim, Vidit Nanda, Steve Smale, and Shmuel Weinberger for many inspirational discussions.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tingran Gao.

Additional information

Editor in Charge: Kenneth Clarkson

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

T. Gao gratefully acknowledges partial support from Simons Math+X Investigators Award 400837, DARPA D15AP00109, NSF IIS 1546413, and an AMS-Simons Travel Grant; J. Brodzki would like to acknowledge the support for this work by the EPSRC grants EP/I016945/1 and EP/N014189/1; S. Mukherjee would like to acknowledge support from NSF DMS 16-13261, NSF IIS 1546331, NSF DMS-1418261, NSF IIS-1320357, NSF DMS-1045153, and HFSP RGP0051/2017.

Appendices

Appendix A: Proofs of Proposition 1.2 and Formula (37)

Proof of Proposition 1.2

The construction of \({\mathfrak {U}}\) using the stars of the vertices of \(\Gamma \) ensures that

  1. (1)

    \(U_i\cap U_j\ne \emptyset \) if and only if \(( i,j )\in E\);

  2. (2)

    \(U_i\cap U_j\cap U_k \ne \emptyset \) if and only if the 2-simplex (ijk) is in X.

For such pair (ij), define constant map \(g_{ij}:U_i\cap U_j\rightarrow G\) as

$$\begin{aligned} g_{ij} (x)=\rho _{ij}\quad \forall x\in U_i\cap U_j. \end{aligned}$$

Set \(g_{ii}=e\) for all \(1\le i\le | V |\), and note that \(g_{ij} (x)=g_{ji}^{-1} (x)\) for all \(x\in U_i\cap U_j\) by our assumption on \(\rho \). If \(\rho \) is synchronizable over G, let \(f:V\rightarrow G\) be a vertex potential satisfying \(\rho \), then \(\rho _{ij}=f_i f_j^{-1}\) for all \(( i,j )\in E\) from (1). Thus \(\rho _{kj}\rho _{ji}=\rho _{ki}\) for any triangle (ijk) in \(\Gamma \), or equivalently that \(g_{kj} ( x )g_{ji} (x)=g_{ki} (x)\) for all \(x\in U_i\cap U_j \cap U_k\). Therefore, \(\{ g_{ij}\,{|}\, 1\le i,j\le | V |\}\) defines a system of coordinate transformations [140, Sect. 2] with values in G. These data determine a principal fibre bundle \({\mathscr {P}}_{\rho }\) with base space \({\mathcal {X}}\) and structure group G—by a standard construction in the theory of fibre bundles (see e.g. [140, Sect. 3.2])—of which local trivializations are defined on the open sets in \({\mathfrak {U}}\) with constant transition functions \(g_{ij}\); this principal bundle is thus flat by definition. Furthermore, the vertex potential f and the compatibility constraints (1) ensure that the following global section \(s:{\mathcal {X}}\rightarrow {\mathscr {P}}_{\rho }\) is well-defined on this bundle:

$$\begin{aligned} s (x)=\phi _i( x, f_i),\quad x\in U_i \end{aligned}$$

where \(\phi _i:U_i\times G\rightarrow {\mathscr {P}}_{\rho }\) is the local trivialization of \({\mathscr {P}}_{\rho }\) over \(U_i\). The triviality of this principal bundle then follows from the existence of such a global section; see e.g. [140, Sect. 8.3]. The other direction of the proposition follows immediately from this triviality criterion for principal bundles. \(\square \)

Proof of Formula (37)

$$\begin{aligned} \langle \omega , \eta \rangle&= \frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F+w_{ji} \big \langle p_j\big (\omega ^{( j )}_{ji}\big ), p_j\big (\eta ^{( j )}_{ji}\big ) \big \rangle _F\Big ]\\&=\frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big )\big \rangle _F+w_{ij} \big \langle \rho _{ij} p_j\big (\omega ^{( j )}_{ji}\big ), \rho _{ij}p_j\big (\eta ^{( j )}_{ji}\big ) \big \rangle _F\Big ]\\&=\frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F+w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ji}\big ), \rho _{ij}p_i\big (\eta ^{( i )}_{ji}\big ) \big \rangle _F\Big ] \\&\quad \quad \quad \quad \quad \quad \quad \quad \text {(see compatibility condition } (31))\\&=\frac{1}{2}\sum _{( i,j )\in E}\Big [w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F+w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), \rho _{ij}p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F\Big ] \\&\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \text {(skew-symmetry)}\\&=\sum _{( i,j )\in E}w_{ij} \big \langle p_i\big (\omega ^{( i )}_{ij}\big ), p_i\big (\eta ^{( i )}_{ij}\big ) \big \rangle _F. \end{aligned}$$

\(\square \)

Appendix B: Graph Laplacian in Discrete Hodge Theory

Define \({\mathbb {K}}\)-valued 0- and 1-forms on weighted graph \(\Gamma =( V,E,w )\) as

$$\begin{aligned} \Omega ^0 ( \Gamma ) := \{ f:V\rightarrow {\mathbb {K}} \},\quad \Omega ^1 ( \Gamma ) := \{ \omega :E\rightarrow {\mathbb {K}}\mid \omega _{ij}=-\omega _{ji}\,\,\forall ( i,j )\in E \}, \end{aligned}$$

equipped with natural inner products

$$\begin{aligned} \begin{aligned} \langle f,g \rangle&:=\sum _id_i \langle f_i, g_i \rangle _{{\mathbb {K}}}, \quad \forall f,g\in \Omega ^0 ( \Gamma ),\\ \langle \omega ,\eta \rangle&:=\sum _{( i,j )\in E}w_{ij}\langle \omega _{ij}, \eta _{ij} \rangle _{{\mathbb {K}}},\quad \forall \omega ,\eta \in \Omega ^1 ( \Gamma ), \end{aligned} \end{aligned}$$

where \(\langle \cdot ,\cdot \rangle _{{\mathbb {K}}}\) is an inner product on \({\mathbb {K}}\), and \(d_i=\sum _{j:( i,j )\in E}w_{ij}\) is the weighted degree at vertex \(i\in V\). Analogous to the study of differential forms on a smooth manifold, one can define the differential \(d:\Omega ^0 ( \Gamma )\rightarrow \Omega ^1 ( \Gamma )\) and codifferential \(\delta :\Omega ^1 ( \Gamma )\rightarrow \Omega ^0 ( \Gamma )\) operators that are formal adjoints of each other:

$$\begin{aligned} (df)_{ij}=f_i-f_j,\quad \forall f\in \Omega ^0 ( \Gamma ),\quad (\delta \omega )_i:=\frac{1}{d_i}\sum _{j:( i,j )\in E}w_{ij}\omega _{ij},\quad \forall \omega \in \Omega ^1 ( \Gamma ). \end{aligned}$$

These constructions can be encoded into a de Rham cochain complex

which realizes \(L_0^{\mathrm {rw}}\), the graph random walk Laplacian, as the Hodge Laplacian of degree zero:

$$\begin{aligned} \big (\Delta ^{( 0 )}f\big )_i:=(\delta df)_i=\frac{1}{d_i}\sum _{j:( i,j )\in E}w_{ij} ( f_i-f_j )=(L_0^{\mathrm {rw}}f)_i,\quad \forall i\in V,\,\,\forall f\in \Omega ^0 ( \Gamma ). \end{aligned}$$

It is well known that \(L_0^{\mathrm {rw}}\) differs from the normalized graph Laplacian \(L_0\) by a similarity transform \(L_0=D^{-1/2}L_0^{\mathrm {rw}}D^{1/2}\), where D is a diagonal matrix with weighted degrees of each vertex on its diagonal.

Software MATLAB code implementing SynCut for the numerical simulations and application in automated geometric morphometrics is publicly available at https://github.com/trgao10/GOS-SynCut.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, T., Brodzki, J. & Mukherjee, S. The Geometry of Synchronization Problems and Learning Group Actions. Discrete Comput Geom 65, 150–211 (2021). https://doi.org/10.1007/s00454-019-00100-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-019-00100-2

Keywords

Mathematics Subject Classification

Navigation