Tính Toán Trung Điểm Riemann trên Các Manifold Hadamard

Karcher, H.: Riemannian Center of Mass and so called Karcher mean, pp. 1–3. arXiv:1407.2087 (2014) Grove, K., Karcher, H.: How to conjugate \(C^1\)-close group actions. Math. Z. 132(1), 11–20 (1973) Kum, S., Sangwoon, Y.: Incremental gradient method for Karcher mean on symmetric cones. J. Optim. Theory Appl. 172, 141–155 (2017) Bini, D.A., Iannazzo, B.: Computing the Karcher mean of symmetric positive definite matrices. Linear Algebra Appl. 438, 1700–1710 (2013) Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 26, 735–747 (2005) Moakher, M.: On the averaging of symmetric positive-definite tensors. J. Elast. 82, 273–296 (2006) Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2008) Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM J. Matrix Anal. Appl. 29, 328–347 (2007) Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. Int. J. Comput. Vis. 66, 41–66 (2006) Lang, S.: Fundamentals of Differential Geometry. Springer, Berlin (1998) Kristály, A., Morosanu, G., Róth, A.: Optimal placement of a deposit between markets: Riemannian–Finsler geometrical approach. J. Optim. Theory Appl. 139(2), 263–276 (2008) Bačák, M.: Computing medians and means in Hadamard spaces. SIAM J. Optim. 24, 1542–1566 (2014) Afsari, B., Tron, R., Vidal, R.: On the convergence of gradient descent for finding the riemannian center of mass. SIAM J. Control Optim. 51, 2230–2260 (2013) Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002) Hosseini, S., Pouryayevali, M.R.: On the metric projection onto prox-regular subsets of Riemannian manifolds. Proc. Am. Math. Soc. 141, 233–244 (2013) Kristály, A.: Nash-type equilibria on Riemannian manifolds: a variational approach. J. Math. Pures Appl. 101(5), 660–688 (2014) Li, C., Mordukhovich, B.S., Wang, J., Yao, J.C.: Weak sharp minima on Riemannian manifolds. SIAM J. Optim. 21, 1523–1560 (2011) Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009) Li, C., Yao, J.C.: Variational inequalities for set-valued vector fields on Riemannian manifolds: convexity of the solution set and the proximal point algorithm. SIAM J. Control Optim. 50, 2486–2514 (2012) Németh, S.Z.: Variational inequalities on Hadamard manifolds. Nonlinear Anal. Theory Methods Appl. 52(5), 1491–1498 (2003) Wang, X., Li, C., Wang, J., Yao, J.C.: Linear convergence of subgradient algorithm for convex feasibility on Riemannian manifolds. SIAM J. Optim. 25, 2334–2358 (2015) Luenberger, D.G.: The gradient projection method along geodesics. Manag. Sci. 18, 620–631 (1972) Gabay, D.: Minimizing a differentiable function over a differentiable manifold. J. Optim. Theory Appl. 37, 177–219 (1982) Udriste, C.: Convex functions and optimization algorithms on Riemannian manifolds. In: Mathematics and its Applications, vol. 297. Kluwer Academic Publishers, Dordrecht (1994) Smith, S.T.: Optimization Techniques on Riemannian Manifolds, vol. 3, pp. 113–146. Fields Institute Communications, American Mathematical Society, Providence (1994) Rapcsák, T.: Smooth Nonlinear Optimization in \(\mathbb{R}^n\). Kluwer Academic Publishers, Dordrecht (1997) Cruz Neto, J.X., Lima, L.L., Oliveira, P.R.: Geodesic algorithms in riemannian geometry. Balkan J. Geom. Appl. 3, 89–100 (1998) Absil, P.-A., Mahony, R., Andrews, B.: Convergence of the iterates of descent methods for analytic cost functions. SIAM J. Optim. 16, 531–547 (2005) Attouch, H., Bolte, J., Svaiter, B.F.: Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss–Seidel methods. Math. Program. Ser. A 137, 91–129 (2013) Lageman, C.: Convergence of gradient-like dynamical systems and optimization algorithms. Ph.D. thesis, Universitat Wurzburg (2007) Łojasiewicz, S.: Une propriété topologique des sous-ensembles analytiques réels. Les Équations aux Dérivées Partielles, Éditions du centre National de la Recherche Scientifique, 87–89 (1963) Kurdyka, K.: On gradients of functions definable in o-minimal structures. Ann. Inst. Fourier 48, 769–783 (1998) Bolte, J., Danillidis, J.A., Lewis, A.: The Lojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems. SIAM J. Optim. 17, 1205–1223 (2006) Attouch, H., Bolte, J.: On the convergence of the proximal algorithm for nonsmooth functions involving analytic features. Math. Program. Ser. B 116(1–2), 5–16 (2009) Bolte, J., Danillidis, J.A., Lewis, A., Shiota, M.: Clarke subgradients of stratifiable functions. SIAM J. Optim. 18, 556–572 (2007) Kurdyka, K., Mostowki, T., Parusinki, A.: Proof of the gradient conjecture of R. Thom. Ann. Math. 152, 763–792 (2000) Bento, G. C., Cruz Neto, J. X., Oliveira, P. R.: Convergence of inexact descent methods for nonconvex optimization on Riemannian manifolds, pp. 1–28. arXiv:1103.4828v1 (2011) Schneider, R., Uschmajew, A.: Convergence Results for projected line-search methods on varieties of low-rank matrices via Łojasiewicz inequality. SIAM J. Optim. 25(1), 622–646 (2015) Cruz Neto, J.X., Oliveira, P.R., Soares Jr., P.A., Soubeyran, A.: Learning how to play Nash, potential games and alternating minimization method for structured nonconvex problems on riemannian manifolds. J. Convex Anal. 20, 395–438 (2013) Hirch, M.W.: Differential Topology. Spring, New York (1976) Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs. American Mathematical Society, Providence (1996) Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Contributions to the study of monotone vector fields. Acta Math. Hung. 94, 307–320 (2002) Rothaus, O.S.: Domains of positivity. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 24, 189–235 (1960) Nesterov, Y.E., Todd, M.J.: On the Riemannian geometry defined by self-concordant barriers and interior-point methods. Found. Comput. Math. 2, 333–361 (2002) Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on the manifold of multivariate normal distributions: theory and application to diffusion tensor MRI processing. J. Math. Imaging Vis. 25, 423–444 (2006)