A Riemannian geometric framework for manifold learning of non-Euclidean data. (English) Zbl 07433034

Jang, Cheongjae; Noh, Yung-Kyun; Park, Frank Chongwoo

Adv. Data Anal. Classif., ADAC 15, No. 3, 673-699 (2021).

Summary: A growing number of problems in data analysis and classification involve data that are non-Euclidean. For such problems, a naive application of vector space analysis algorithms will produce results that depend on the choice of local coordinates used to parametrize the data. At the same time, many data analysis and classification problems eventually reduce to an optimization, in which the criteria being minimized can be interpreted as the distortion associated with a mapping between two curved spaces. Exploiting this distortion minimizing perspective, we first show that manifold learning problems involving non-Euclidean data can be naturally framed as seeking a mapping between two Riemannian manifolds that is closest to being an isometry. A family of coordinate-invariant first-order distortion measures is then proposed that measure the proximity of the mapping to an isometry, and applied to manifold learning for non-Euclidean data sets. Case studies ranging from synthetic data to human mass-shape data demonstrate the many performance advantages of our Riemannian distortion minimization framework.

MSC:

53A35	Non-Euclidean differential geometry
53B21	Methods of local Riemannian geometry
58C35	Integration on manifolds; measures on manifolds
58E20	Harmonic maps, etc.

Keywords:

manifold learning; non-euclidean data; Riemannian geometry; distortion; harmonic map

References:

[1]	Barahona, S.; Gual-Arnau, X.; Ibáñez, MV; Simó, A., Unsupervised classification of children’s bodies using currents, Adv Data Anal Classif, 12, 2, 365-397 (2018) · Zbl 1414.62431
[2]	Belkin, M.; Niyogi, P., Laplacian eigenmaps for dimensionality reduction and data representation, Neural Comput, 15, 6, 1373-1396 (2003) · Zbl 1085.68119
[3]	Belkin, M.; Niyogi, P.; Sindhwani, V., Manifold regularization: a geometric framework for learning from labeled and unlabeled examples, J Mach Learn Res, 7, 2399-2434 (2006) · Zbl 1222.68144
[4]	Boothby, WM, An introduction to differentiable manifolds and Riemannian geometry (1986), Cambridge: Academic press, Cambridge · Zbl 0596.53001
[5]	Bronstein, MM; Bruna, J.; LeCun, Y.; Szlam, A.; Vandergheynst, P., Geometric deep learning: going beyond euclidean data, IEEE Signal Process Magazine, 34, 4, 18-42 (2017)
[6]	Coifman, RR; Lafon, S., Diffusion maps, Appl Comput Harmonic Anal, 21, 1, 5-30 (2006) · Zbl 1095.68094
[7]	Desbrun, M.; Meyer, M.; Alliez, P., Intrinsic parameterizations of surface meshes, Comput Graph Forum Wiley Online Libr, 21, 209-218 (2002)
[8]	Donoho, DL; Grimes, C., Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data, Proc Natl Acad Sci, 100, 10, 5591-5596 (2003) · Zbl 1130.62337
[9]	Dubrovin, BA; Fomenko, AT; Novikov, SP, Modern geometry-methods and applications Part I. The geometry of surfaces, transformation groups, and fields (1992), Berlin: Springer, Berlin · Zbl 0751.53001
[10]	Eells, J.; Lemaire, L., A report on harmonic maps, Bull London Math Soc, 10, 1, 1-68 (1978) · Zbl 0401.58003
[11]	Eells, J.; Lemaire, L., Another report on harmonic maps, Bull London Math Soc, 20, 5, 385-524 (1988) · Zbl 0669.58009
[12]	Eells, J.; Sampson, JH, Harmonic mappings of Riemannian manifolds, Am J Math, 86, 1, 109-160 (1964) · Zbl 0122.40102
[13]	Feragen A, Lauze F, Hauberg S (2015) Geodesic exponential kernels: When curvature and linearity conflict. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp 3032-3042
[14]	Fletcher, PT; Joshi, S., Riemannian geometry for the statistical analysis of diffusion tensor data, Signal Process, 87, 2, 250-262 (2007) · Zbl 1186.94126
[15]	Goldberg, Y.; Zakai, A.; Kushnir, D.; Ritov, Y., Manifold learning: the price of normalization, J Mach Learn Res, 9, 1909-1939 (2008) · Zbl 1225.68181
[16]	Gu, X.; Wang, Y.; Chan, TF; Thompson, PM; Yau, ST, Genus zero surface conformal mapping and its application to brain surface mapping, IEEE Trans Med Imag, 23, 8, 949-958 (2004)
[17]	Jang C (2019) Riemannian distortion measures for non-euclidean data. Ph.D. thesis, Seoul National University
[18]	Jayasumana, S.; Hartley, R.; Salzmann, M.; Li, H.; Harandi, M., Kernel methods on riemannian manifolds with gaussian rbf kernels, IEEE Trans Pattern Anal Mach Intell, 37, 12, 2464-2477 (2015)
[19]	Lafon SS (2004) Diffusion maps and geometric harmonics. PhD thesis, Yale University Ph.D dissertation
[20]	Lee, T.; Park, FC, A geometric algorithm for robust multibody inertial parameter identification, IEEE Robot Autom Lett, 3, 3, 2455-2462 (2018)
[21]	Lin, B.; He, X.; Ye, J., A geometric viewpoint of manifold learning, Appl Inform, 2, 3 (2015)
[22]	McQueen J, Meila M, Perrault-Joncas D (2016) Nearly isometric embedding by relaxation. In: Lee D, Sugiyama M, Luxburg U, Guyon I, Garnett R (eds) Advances in Neural Information Processing Systems, pp 2631-2639
[23]	Mullen, P.; Tong, Y.; Alliez, P.; Desbrun, M., Spectral conformal parameterization, Comput Graph Forum Wiley Online Libr, 27, 1487-1494 (2008)
[24]	Park, FC; Brockett, RW, Kinematic dexterity of robotic mechanisms, Int J Robot Res, 13, 1, 1-15 (1994)
[25]	Pelletier, B., Kernel density estimation on riemannian manifolds, Stat Probab Lett, 73, 3, 297-304 (2005) · Zbl 1065.62063
[26]	Perrault-Joncas D, Meila M (2013) Non-linear dimensionality reduction: Riemannian metric estimation and the problem of geometric discovery. arXiv preprint arXiv:1305.7255
[27]	Roweis, ST; Saul, LK, Nonlinear dimensionality reduction by locally linear embedding, Science, 290, 5500, 2323-2326 (2000)
[28]	Steinke, F.; Hein, M.; Schölkopf, B., Nonparametric regression between general riemannian manifolds, SIAM J Imag Sci, 3, 3, 527-563 (2010) · Zbl 1195.41011
[29]	Tenenbaum, JB; De Silva, V.; Langford, JC, A global geometric framework for nonlinear dimensionality reduction, Science, 290, 5500, 2319-2323 (2000)
[30]	Vinué, G.; Simó, A.; Alemany, S., The \(k\)-means algorithm for 3d shapes with an application to apparel design, Adv Data Anal Classif, 10, 1, 103-132 (2016) · Zbl 1414.62295
[31]	Wensing, PM; Kim, S.; Slotine, JJE, Linear matrix inequalities for physically consistent inertial parameter identification: a statistical perspective on the mass distribution, IEEE Robot Autom Lett, 3, 1, 60-67 (2018)
[32]	Yang Y, Yu Y, Zhou Y, Du S, Davis J, Yang R (2014) Semantic parametric reshaping of human body models. In: 3D Vision (3DV), 2014 2nd International Conference on, IEEE, vol 2, pp 41-48
[33]	Zhang, T.; Li, X.; Tao, D.; Yang, J., Local coordinates alignment (lca): a novel manifold learning approach, Int J Pattern Recogn Artif Intell, 22, 4, 667-690 (2008)
[34]	Zhang, Z.; Zha, H., Principal manifolds and nonlinear dimensionality reduction via tangent space alignment, SIAM J Sci Comput, 26, 1, 313-338 (2004) · Zbl 1077.65042