In many applications, the curvature of the space supporting the data makes the statistical modeling challenging. In this paper we discuss the construction and use of probability distributions wrapped around manifolds using exponential maps. These distributions have already been used on specific manifolds. We describe their construction in the unifying framework of affine locally symmetric spaces. Affine locally symmetric spaces are a broad class of manifolds containing many manifolds encountered in the data sciences. We show that on these spaces, exponential-wrapped distributions enjoy interesting properties for practical use. We provide the generic expression of the Jacobian appearing in these distributions and compute it on two particular examples: Grassmannians and pseudohyperboloids. We illustrate the interest of such distributions in a classification experiment on simulated data.


  1. statistics on manifolds
  2. exponential map
  3. differential of the exponential map
  4. affine locally symmetric spaces
  5. Lie groups
  6. wrapped distributions

MSC codes

  1. 53Z50
  2. 62E99
  3. 53C35

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E. Chevallier, F. Barbaresco, and J. Angulo (2015), Probability density estimation on the hyperbolic space applied to radar processing, in International Conference on Geometric Science of Information, Springer, Cham, Switzerland, pp. 753–761.
E. Chevallier, T. Forget, F. Barbaresco, and J. Angulo (2016), Kernel density estimation on the Siegel space with an application to radar processing, Entropy, 18, 396.
E. Chevallier, E. Kalunga, and J. Angulo (2017), Kernel density estimation on spaces of Gaussian distributions and symmetric positive definite matrices, SIAM J. Imaging Sci., 10, pp. 191–215.
J. Ding and A. Regev (2020), Deep generative model embedding of single-cell RNA-Seq profiles on hyperspheres and hyperbolic spaces, Nature Commun., 12, pp. 1–17.
L. Falorsi, P. de Haan, T. R. Davidson, and P. Forré (2019), Reparameterizing distributions on Lie groups, Proc. Mach. Learn. Res. (PMLR), 89, pp. 3244–3253.
R. A. Fisher (1953), Dispersion on a sphere, Proc. R. Soc. Lond. A Math. Phys. Sci. Eng., 217, pp. 295–305.
P. T. Fletcher, S. Joshi, C. Lu, and S. Pizer (2003), Gaussian distributions on Lie groups and their application to statistical shape analysis, in Biennial International Conference on Information Processing in Medical Imaging, Springer, Berlin, pp. 450–462.
A. V. Gavrilov (2007), The double exponential map and covariant derivation, Sib. Math. J., 48, pp. 56–61.
P. Hall, G. Watson, and J. Cabrera (1987), Kernel density estimation with spherical data, Biometrika, 74, pp. 751–762.
S. Hauberg (2018), Directional statistics with the spherical normal distribution, in Proceedings of the 21st International Conference on Information Fusion (FUSION), IEEE, Piscataway, NJ, pp. 704–711.
S. Helgason (1979), Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York.
H. Hendriks (1990), Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions, Ann. Statist., pp. 832–849.
S. F. Huckemann, P. T. Kim, J.-Y. Koo, and A. Munk (2010), Möbius deconvolution on the hyperbolic plane with application to impedance density estimation, Ann. Stat., 38, pp. 2465–2498.
G. Jona-Lasino, A. Gelfand, and M. Jona-Lasino (2012), Spatial analysis of wave directional data using wrapped Gaussian processes, Ann. Appl. Stat., 6, pp. 1478–1498.
S. Kato and P. McCullagh (2020), Some properties of a Cauchy family on the sphere derived from the Möbius transformations, Bernoulli, 26, pp. 3224–3248.
P. T. Kim (1998), Deconvolution density estimation on SO(N), Ann. Statist., 26, pp. 1083–1102.
S. Kurtek, A. Srivastava, E. Klassen, and Z. Ding (2012), Statistical modeling of curves using shapes and related features, J. Amer. Statist. Assoc., 107, pp. 1152–1165.
M. T. Law and J. Stam (2020), Ultrahyperbolic Representation Learning, in Adv. Neural Inf. Process. Syst. 34, Curran Associates, Red Hook, NY, pp. 1668–1678.
L.-H. Lim, K. S.-W. Wong, and K. Ye (2021), The Grassmannian of affine subspaces, Found. Comput. Math., 21, pp. 537–574.
A. Mallasto and A. Feragen (2018), Wrapped Gaussian process regression on Riemannian manifolds, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE Computer Society, Los Alamitos, CA, pp. 5580–5588.
A. Mallasto, S. Hauberg, and A. Feragen (2019), Probabilistic Riemannian submanifold learning with wrapped Gaussian process latent variable models, Proc. Mach. Learn. Res. (PMLR), 89, pp. 2368–2377.
K. V. Mardia (1972), Statistics of Directional Data, Academic Press, London.
E. Mathieu, C. Le Lan, C. J. Maddison, R. Tomioka, and Y. W. Teh (2019), Continuous hierarchical representations with Poincaré variational auto-encoders, in Advances in Neural Information Processing Systems, Curran Associates, Red Hook, NY, pp. 12544–12555.
N. Miolane, N. Guigui, A. L. Brigant, J. Mathe, B. Hou, Y. Thanwerdas, S. Heyder, O. Peltre, N. Koep, H. Zaatiti, H. Hajri, Y. Cabanes, T. Gerald, P. Chauchat, C. Shewmake, D. Brooks, B. Kainz, C. Donnat, S. Holmes, and X. Pennec (2020), Geomstats: A Python package for Riemannian geometry in machine learning, J. Mach. Learn. Res., 21, pp. 1–9, http://jmlr.org/papers/v21/19-027.html.
E. Nava-Yazdani, H.-C. Hege, T. J. Sullivan, and C. von Tycowicz (2020), Geodesic analysis in Kendall’s shape space with epidemiological applications, J. Math. Imaging Vision, 62, pp. 549–559.
K. Nomizu (1954), Invariant affine connections on homogeneous spaces, Amer. J. Math., 76, pp. 33–65.
B. Pelletier (2005), Kernel density estimation on Riemannian manifolds, Statist. Probab. Lett., 73, pp. 297–304.
X. Pennec (2006), Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measurements, J. Math. Imaging Vision, 25, pp. 127–154.
X. Pennec (2019), Curvature Effects on the Empirical Mean in Riemannian and Affine Manifolds: A Non-Asymptotic High Concentration Expansion in the Small-Sample Regime, preprint, arXiv:1906.07418.
X. Pennec and M. Lorenzi (2020), Beyond Riemannian geometry: The affine connection setting for transformation groups, in Riemannian Geometric Statistics in Medical Image Analysis, Academic Press, San Diego, CA, pp. 169–229.
S. Said, L. Bombrun, Y. Berthoumieu, and J. H. Manton (2017a), Riemannian Gaussian distributions on the space of symmetric positive definite matrices, IEEE Trans. Inform. Theory, 63, pp. 2153–2170.
S. Said, H. Hajri, L. Bombrun, and B. C. Vemuri (2017b), Gaussian distributions on Riemannian symmetric spaces: Statistical learning with structured covariance matrices, IEEE Trans. Inform. Theory, 64, pp. 752–772.
R. Slama, H. Wannous, and M. Daoudi (2014), Grassmannian representation of motion depth for 3D human gesture and action recognition, in 22nd International Conference on Pattern Recognition, IEEE, Piscataway, NJ, pp. 3499–3504.
R. Slama, H. Wannous, M. Daoudi, and A. Srivastava (2015), Accurate 3D action recognition using learning on the Grassmann manifold, Pattern Recognit., 48, pp. 556–567.
A. Srivastava, S. Joshi, W. Mio, and X. Liu (2005), Statistical shape analysis: Clustering, learning, and testing, Trans. Pattern Anal. Mach. Intell., 27, pp. 590–602.
H. Taniguchi (1984), A note on the differential of the exponential map and Jacobi fields in a symmetric space, Tokyo J. Math., 7, pp. 177–181.
A. Terras (1988), Harmonic Analysis on Symmetric Spaces and Applications II, Berlin, Springer.
P. Turaga, A. Veeraraghavan, A. Srivastava, and R. Chellappa (2011), Statistical computations on Grassmann and Stiefel manifolds for image and video-based recognition, Trans. Pattern Anal. Mach. Intell., 33, pp. 2273–2286.

Information & Authors


Published In

cover image SIAM Journal on Mathematics of Data Science
SIAM Journal on Mathematics of Data Science
Pages: 1347 - 1368
ISSN (online): 2577-0187


Submitted: 30 November 2021
Accepted: 3 October 2022
Published online: 20 December 2022


  1. statistics on manifolds
  2. exponential map
  3. differential of the exponential map
  4. affine locally symmetric spaces
  5. Lie groups
  6. wrapped distributions

MSC codes

  1. 53Z50
  2. 62E99
  3. 53C35



Emmanuel Chevallier Contact the author
Aix Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, Marseille, France.
Didong Li
Aix Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, Marseille, France.
Yulong Lu
Aix Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, Marseille, France.
David Dunson
Aix Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, Marseille, France.
Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA.
Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA, USA.
Department of Statistical Sciences and Mathematics, Duke University, Durham, NC, USA.

Funding Information

Office of Naval Research (ONR): N00014-14-1-0245, N00014-16-1-2147
Funding. The authors acknowledge support for this research from an Office of Naval Research grants N00014-14-1-0245 and N00014-16-1-2147. The third author is supported by the US National Science Foundation under award DMS-2107934.

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