Abstract.

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.

Keywords

  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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Information & Authors

Information

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

History

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

Keywords

  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

Authors

Affiliations

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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