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Computing Riemannian Center of Mass on Hadamard Manifolds

Author

Listed:
  • Glaydston Carvalho Bento

    (Universidade Federal de Goiás)

  • Sandro Dimy Barbosa Bitar

    (Universidade Federal do Amazonas)

  • João Xavier Cruz Neto

    (Universidade Federal do Piauí)

  • Paulo Roberto Oliveira

    (Universidade Federal do Rio de Janeiro)

  • João Carlos Oliveira Souza

    (Universidade Federal do Piauí)

Abstract

In this paper, we perform the steepest descent method for computing Riemannian center of mass on Hadamard manifolds. To this end, we extend convergence of the method to the Hadamard setting for continuously differentiable (possible nonconvex) functions which satisfy the Kurdyka–Łojasiewicz property. Some numerical experiments computing $$L^1$$ L 1 and $$L^2$$ L 2 center of mass in the context of positive definite symmetric matrices are presented using two different stepsize rules.

Suggested Citation

  • Glaydston Carvalho Bento & Sandro Dimy Barbosa Bitar & João Xavier Cruz Neto & Paulo Roberto Oliveira & João Carlos Oliveira Souza, 2019. "Computing Riemannian Center of Mass on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 183(3), pages 977-992, December.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:3:d:10.1007_s10957-019-01580-1
    DOI: 10.1007/s10957-019-01580-1
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    References listed on IDEAS

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    1. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    2. Sangho Kum & Sangwoon Yun, 2017. "Incremental Gradient Method for Karcher Mean on Symmetric Cones," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 141-155, January.
    3. NESTEROV , Yu. & TODD, Mike, 2002. "On the Riemannian geometry defined by self-concordant barriers and interior-point methods," LIDAM Reprints CORE 1595, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Joao Xavier Cruz Neto & Paulo Roberto Oliveira & A. Soares Jr Pedro & Antoine Soubeyran, 2013. "Learning how to Play Nash, Potential Games and Alternating Minimization Method for Structured Nonconvex Problems on Riemannian Manifolds," Post-Print hal-01500875, HAL.
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    Cited by:

    1. Erik Alex Papa Quiroz & Nancy Baygorrea Cusihuallpa & Nelson Maculan, 2020. "Inexact Proximal Point Methods for Multiobjective Quasiconvex Minimization on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 879-898, September.

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