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Статті в журналах з теми "Convergence of Riemannian manifolds"

Автор: Grafiati

Опубліковано: 10 березня 2023

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1

Perales, Raquel. "Convergence of manifolds and metric spaces with boundary." Journal of Topology and Analysis 12, no. 03 (November 28, 2018): 735–74. http://dx.doi.org/10.1142/s1793525319500638.

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We study sequences of oriented Riemannian manifolds with boundary and, more generally, integral current spaces and metric spaces with boundary. We prove theorems demonstrating when the Gromov–Hausdorff (GH) and Sormani–Wenger Intrinsic Flat (SWIF) limits of sequences of such metric spaces agree. Thus in particular the limit spaces are countably [Formula: see text] rectifiable spaces. From these theorems we derive compactness theorems for sequences of Riemannian manifolds with boundary where both the GH and SWIF limits agree. For sequences of Riemannian manifolds with boundary we only require non-negative Ricci curvature, upper bounds on volume, noncollapsing conditions on the interior of the manifold and diameter controls on the level sets near the boundary.

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2

Kasue, Atsushi, and Hironori Kumura. "Spectral convergence of Riemannian manifolds." Tohoku Mathematical Journal 46, no. 2 (1994): 147–79. http://dx.doi.org/10.2748/tmj/1178225756.

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3

Greene, Robert, and Hung-Hsi Wu. "Lipschitz convergence of Riemannian manifolds." Pacific Journal of Mathematics 131, no. 1 (January 1, 1988): 119–41. http://dx.doi.org/10.2140/pjm.1988.131.119.

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4

Argyros, Ioannis K., and Santhosh George. "ON THE SEMILOCAL CONVERGENCE OF NEWTON'S METHOD FOR SECTIONS ON RIEMANNIAN MANIFOLDS." Asian-European Journal of Mathematics 07, no. 01 (March 2014): 1450007. http://dx.doi.org/10.1142/s1793557114500077.

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We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] in combination with the weaker center 2-piece L1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal.22 (2002) 359–390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math.8 (2008) 197–226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant α-theory, IMA J. Numer. Anal.23 (2003) 395–419].

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5

Yang, Le. "Riemannian median and its estimation." LMS Journal of Computation and Mathematics 13 (December 2010): 461–79. http://dx.doi.org/10.1112/s1461157020090531.

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AbstractIn this paper, we define the geometric median for a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to compute the geometric median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also improves a recent result of P. T. Fletcheret al. [NeuroImage45 (2009) S143–S152].

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6

Kasue, Atsushi. "A convergence theorem for Riemannian manifolds and some applications." Nagoya Mathematical Journal 114 (June 1989): 21–51. http://dx.doi.org/10.1017/s0027763000001380.

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Анотація:

The purpose of the present paper is first to reformulate a Lipschitz convergence theorem for Riemannian manifolds originally introduced by Gromov [17] and secondly to give some applications of the theorem to a class of open Riemannian manifolds.

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7

Kasue, Atsushi, and Hironori Kumura. "Spectral convergence of Riemannian manifolds, II." Tohoku Mathematical Journal 48, no. 1 (1996): 71–120. http://dx.doi.org/10.2748/tmj/1178225413.

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8

Boumal, Nicolas, P.-A. Absil, and Coralia Cartis. "Global rates of convergence for nonconvex optimization on manifolds." IMA Journal of Numerical Analysis 39, no. 1 (February 7, 2018): 1–33. http://dx.doi.org/10.1093/imanum/drx080.

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Abstract We consider the minimization of a cost function f on a manifold $\mathcal{M}$ using Riemannian gradient descent and Riemannian trust regions (RTR). We focus on satisfying necessary optimality conditions within a tolerance ε. Specifically, we show that, under Lipschitz-type assumptions on the pullbacks of f to the tangent spaces of $\mathcal{M}$, both of these algorithms produce points with Riemannian gradient smaller than ε in $\mathcal{O}\big(1/\varepsilon ^{2}\big)$ iterations. Furthermore, RTR returns a point where also the Riemannian Hessian’s least eigenvalue is larger than −ε in $\mathcal{O} \big(1/\varepsilon ^{3}\big)$ iterations. There are no assumptions on initialization. The rates match their (sharp) unconstrained counterparts as a function of the accuracy ε (up to constants) and hence are sharp in that sense. These are the first deterministic results for global rates of convergence to approximate first- and second-order Karush-Kuhn-Tucker points on manifolds. They apply in particular for optimization constrained to compact submanifolds of ${\mathbb{R}^{n}}$, under simpler assumptions.

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9

Katsuda, Atsushi. "Gromov’s convergence theorem and its application." Nagoya Mathematical Journal 100 (December 1985): 11–48. http://dx.doi.org/10.1017/s0027763000000209.

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Анотація:

One of the basic questions of Riemannian geometry is that “If two Riemannian manifolds are similar with respect to the Riemannian invariants, for example, the curvature, the volume, the first eigenvalue of the Laplacian, then are they topologically similar?”. Initiated by H. Rauch, many works are developed to the above question. Recently M. Gromov showed a remarkable theorem ([7] 8.25, 8.28), which may be useful not only for the above question but also beyond the above. But it seems to the author that his proof is heuristic and it contains some gaps (for these, see § 1), so we give a detailed proof of 8.25 in [7]. This is the first purpose of this paper. Second purpose is to prove a differentiable sphere theorem for manifolds of positive Ricci curvature, using the above theorem as a main tool.

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10

Greene, Robert, and Hung-Hsi Wu. "Addendum to: “Lipschitz convergence of Riemannian manifolds”." Pacific Journal of Mathematics 140, no. 2 (December 1, 1989): 398. http://dx.doi.org/10.2140/pjm.1989.140.398.

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11

Allen, Brian, and Edward Bryden. "Sobolev bounds and convergence of Riemannian manifolds." Nonlinear Analysis 185 (August 2019): 142–69. http://dx.doi.org/10.1016/j.na.2019.03.001.

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12

Kasue, Atsushi. "Convergence of Riemannian manifolds and Laplace operators. I." Annales de l’institut Fourier 52, no. 4 (2002): 1219–57. http://dx.doi.org/10.5802/aif.1916.

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13

Kasue, Atsushi. "Convergence of Riemannian Manifolds and Laplace Operators, II." Potential Analysis 24, no. 2 (March 2006): 137–94. http://dx.doi.org/10.1007/s11118-005-8568-x.

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14

Wang, Xing, and Chunjie Zhang. "Pointwise Convergence of Solutions to the Schrödinger Equation on Manifolds." Canadian Journal of Mathematics 71, no. 4 (January 7, 2019): 983–95. http://dx.doi.org/10.4153/cjm-2018-001-4.

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AbstractLet $(M^{n},g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity required so that the solution to a free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^{\unicode[STIX]{x1D6FC}}(M)$. For hyperbolic space, the standard sphere, and the two-dimensional torus, we prove that $\unicode[STIX]{x1D6FC}>\frac{1}{2}$ is enough. For general compact manifolds, due to the lack of a local smoothing effect, it is hard to improve on the bound $\unicode[STIX]{x1D6FC}>1$ from interpolation. We managed to go below 1 for dimension ${\leqslant}$ 3. The more interesting thing is that, for a one-dimensional compact manifold, $\unicode[STIX]{x1D6FC}>\frac{1}{3}$ is sufficient.

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15

Argyros, Ioannis K., and Santhosh George. "EXTENDING THE APPLICABILITY OF NEWTON'S METHOD ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE." Asian-European Journal of Mathematics 06, no. 03 (September 2013): 1350041. http://dx.doi.org/10.1142/s1793557113500411.

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We present a new semilocal convergence analysis of Newton's method on Riemannian manifolds with values in a cone in order to solve the inclusion problem. Using more precise majorizing sequences than in earlier studies such as [J. H. Wang, S. Huang and C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwanese J. Math.13(2B) (2009) 633–656] and the concept of L-average Lipschitz condition we provide: weaker sufficient convergence conditions; tighter error analysis on the distances involved and an at least as precise information on the solutions. These advantages are obtained using the same parameters and functions. Applications include the celebrated Newton–Kantorovich theorem.

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16

Wang, J. H. "Convergence of Newton’s Method for Sections on Riemannian Manifolds." Journal of Optimization Theory and Applications 148, no. 1 (August 27, 2010): 125–45. http://dx.doi.org/10.1007/s10957-010-9748-4.

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17

Ferrand, Jacqueline. "Convergence and degeneracy of quasiconformal maps of Riemannian manifolds." Journal d'Analyse Mathématique 69, no. 1 (December 1996): 1–24. http://dx.doi.org/10.1007/bf02787099.

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18

Gordina, Maria, and Thomas Laetsch. "A convergence to Brownian motion on sub-Riemannian manifolds." Transactions of the American Mathematical Society 369, no. 9 (March 1, 2017): 6263–78. http://dx.doi.org/10.1090/tran/6831.

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19

Bl�mlinger, Martin. "Asymptotic distribution and weak convergence on compact Riemannian manifolds." Monatshefte f�r Mathematik 110, no. 3-4 (September 1990): 177–88. http://dx.doi.org/10.1007/bf01301674.

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20

Zhou, Jialun, and Salem Said. "Fast, Asymptotically Efficient, Recursive Estimation in a Riemannian Manifold." Entropy 21, no. 10 (October 21, 2019): 1021. http://dx.doi.org/10.3390/e21101021.

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Stochastic optimisation in Riemannian manifolds, especially the Riemannian stochastic gradient method, has attracted much recent attention. The present work applies stochastic optimisation to the task of recursive estimation of a statistical parameter which belongs to a Riemannian manifold. Roughly, this task amounts to stochastic minimisation of a statistical divergence function. The following problem is considered: how to obtain fast, asymptotically efficient, recursive estimates, using a Riemannian stochastic optimisation algorithm with decreasing step sizes. In solving this problem, several original results are introduced. First, without any convexity assumptions on the divergence function, we proved that, with an adequate choice of step sizes, the algorithm computes recursive estimates which achieve a fast non-asymptotic rate of convergence. Second, the asymptotic normality of these recursive estimates is proved by employing a novel linearisation technique. Third, it is proved that, when the Fisher information metric is used to guide the algorithm, these recursive estimates achieve an optimal asymptotic rate of convergence, in the sense that they become asymptotically efficient. These results, while relatively familiar in the Euclidean context, are here formulated and proved for the first time in the Riemannian context. In addition, they are illustrated with a numerical application to the recursive estimation of elliptically contoured distributions.

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21

Kupferman, Raz, and Cy Maor. "Riemannian surfaces with torsion as homogenization limits of locally Euclidean surfaces with dislocation-type singularities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 4 (July 1, 2016): 741–68. http://dx.doi.org/10.1017/s0308210515000773.

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We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, from the 1950s, models continuously distributed edge dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, that the second model can be obtained rigorously as a homogenization limit of the first model as the density of singular edge dislocation tends to infinity.

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22

Fardoun, Ali, and Rachid Regbaoui. "Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 734 (January 1, 2018): 229–64. http://dx.doi.org/10.1515/crelle-2015-0012.

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AbstractWe prove the compactness of solutions of general fourth order elliptic equations which areL^{1}-perturbations of theQ-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of theQ-curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for theQ-curvature problem when the Paneitz operator is positive with trivial kernel.

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23

Halimu, Yeerjiang, Chao Zhou, Qi You, and Jun Sun. "A Quantum-Behaved Particle Swarm Optimization Algorithm on Riemannian Manifolds." Mathematics 10, no. 22 (November 8, 2022): 4168. http://dx.doi.org/10.3390/math10224168.

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The Riemannian manifold optimization algorithms have been widely used in machine learning, computer vision, data mining, and other technical fields. Most of these algorithms are based on the geodesic or the retracement operator and use the classical methods (i.e., the steepest descent method, the conjugate gradient method, the Newton method, etc.) to solve engineering optimization problems. However, they lack the ability to solve non-differentiable mathematical models and ensure global convergence for non-convex manifolds. Considering this issue, this paper proposes a quantum-behaved particle swarm optimization (QPSO) algorithm on Riemannian manifolds named RQPSO. In this algorithm, the quantum-behaved particles are randomly distributed on the manifold surface and iteratively updated during the whole search process. Then, the vector transfer operator is used to translate the guiding vectors, which are not in the same Euclidean space, to the tangent space of the particles. Through the searching of these guiding vectors, we can achieve the retracement and update of points and finally obtain the optimized result. The proposed RQPSO algorithm does not depend on the expression form of a problem and could deal with various engineering technical problems, including both differentiable and non-differentiable ones. To verify the performance of RQPSO experimentally, we compare it with some traditional algorithms on three common matrix manifold optimization problems. The experimental results show that RQPSO has better performance than its competitors in terms of calculation speed and optimization efficiency.

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24

Wu, Jyh-Yang. "Convergence of Riemannian 3-Manifolds Under a Ricci Curvature Bound." American Journal of Mathematics 116, no. 4 (August 1994): 1019. http://dx.doi.org/10.2307/2375008.

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25

Yang, Deane. "Convergence of riemannian manifolds with integral bounds on curvature. I." Annales scientifiques de l'École normale supérieure 25, no. 1 (1992): 77–105. http://dx.doi.org/10.24033/asens.1644.

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26

Yang, Deane. "Convergence of riemannian manifolds with integral bounds on curvature. II." Annales scientifiques de l'École normale supérieure 25, no. 2 (1992): 179–99. http://dx.doi.org/10.24033/asens.1647.

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27

Giesselmann, Jan. "A convergence result for finite volume schemes on Riemannian manifolds." ESAIM: Mathematical Modelling and Numerical Analysis 43, no. 5 (June 12, 2009): 929–55. http://dx.doi.org/10.1051/m2an/2009013.

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28

Thorstensen, Nicolas, and Otmar Scherzer. "Convergence of variational regularization methods for imaging on Riemannian manifolds." Inverse Problems 28, no. 1 (December 19, 2011): 015007. http://dx.doi.org/10.1088/0266-5611/28/1/015007.

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29

Ogura, Yukio. "Weak convergence of laws of stochastic processes on Riemannian manifolds." Probability Theory and Related Fields 119, no. 4 (April 2001): 529–57. http://dx.doi.org/10.1007/pl00008770.

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30

Fernandes, Teles A., Orizon P. Ferreira, and Jinyun Yuan. "On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds." Journal of Optimization Theory and Applications 173, no. 3 (March 28, 2017): 828–43. http://dx.doi.org/10.1007/s10957-017-1107-2.

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31

Fan, Xiaomeng, Yuwei Wu, Zhi Gao, Yunde Jia, and Mehrtash Harandi. "Efficient Riemannian Meta-Optimization by Implicit Differentiation." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 4 (June 28, 2022): 3733–40. http://dx.doi.org/10.1609/aaai.v36i4.20287.

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Анотація:

To solve optimization problems with nonlinear constrains, the recently developed Riemannian meta-optimization methods show promise, which train neural networks as an optimizer to perform optimization on Riemannian manifolds. A key challenge is the heavy computational and memory burdens, because computing the meta-gradient with respect to the optimizer involves a series of time-consuming derivatives, and stores large computation graphs in memory. In this paper, we propose an efficient Riemannian meta-optimization method that decouples the complex computation scheme from the meta-gradient. We derive Riemannian implicit differentiation to compute the meta-gradient by establishing a link between Riemannian optimization and the implicit function theorem. As a result, the updating our optimizer is only related to the final two iterations, which in turn speeds up our method and reduces the memory footprint significantly. We theoretically study the computational load and memory footprint of our method for long optimization trajectories, and conduct an empirical study to demonstrate the benefits of the proposed method. Evaluations of three optimization problems on different Riemannian manifolds show that our method achieves state-of-the-art performance in terms of the convergence speed and the quality of optima.

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32

Wang, Xiangmei, Chong Li, Jinhua Wang, and Jen-Chih Yao. "Linear Convergence of Subgradient Algorithm for Convex Feasibility on Riemannian Manifolds." SIAM Journal on Optimization 25, no. 4 (January 2015): 2334–58. http://dx.doi.org/10.1137/14099961x.

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33

Argyros, Ioannis K. "An improved unifying convergence analysis of Newton’s method in Riemannian manifolds." Journal of Applied Mathematics and Computing 25, no. 1-2 (September 2007): 345–51. http://dx.doi.org/10.1007/bf02832359.

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34

Hüning, Svenja, and Johannes Wallner. "Convergence of subdivision schemes on Riemannian manifolds with nonpositive sectional curvature." Advances in Computational Mathematics 45, no. 3 (May 2, 2019): 1689–709. http://dx.doi.org/10.1007/s10444-019-09693-x.

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35

Alvarez, F., J. Bolte, and J. Munier. "A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds." Foundations of Computational Mathematics 8, no. 2 (August 3, 2006): 197–226. http://dx.doi.org/10.1007/s10208-006-0221-6.

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36

Ehler, Martin, Manuel Gräf, and Chris J. Oates. "Optimal Monte Carlo integration on closed manifolds." Statistics and Computing 29, no. 6 (October 30, 2019): 1203–14. http://dx.doi.org/10.1007/s11222-019-09894-w.

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Abstract The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $$n^{-1/2}$$n-1/2. However, the re-weighting of random points, as exemplified in the Bayesian Monte Carlo method, can sometimes be used to improve the convergence order. This paper contributes general theoretical results for Sobolev spaces on closed Riemannian manifolds, where we verify that such re-weighting yields optimal approximation rates up to a logarithmic factor. We also provide numerical experiments matching the theoretical results for some Sobolev spaces on the sphere $${\mathbb {S}}^2$$S2 and on the Grassmannian manifold $${\mathcal {G}}_{2,4}$$G2,4. Our theoretical findings also cover function spaces on more general sets such as the unit ball, the cube, and the simplex.

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37

Pisante, Adriano, and Fabio Punzo. "Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows." Communications in Contemporary Mathematics 17, no. 05 (October 2015): 1450041. http://dx.doi.org/10.1142/s0219199714500412.

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Анотація:

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

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38

Wang, Jinhua, Xiangmei Wang, Chong Li, and Jen-Chih Yao. "Convergence Analysis of Gradient Algorithms on Riemannian Manifolds without Curvature Constraints and Application to Riemannian Mass." SIAM Journal on Optimization 31, no. 1 (January 2021): 172–99. http://dx.doi.org/10.1137/19m1289285.

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39

Duan, Xiaomin, Huafei Sun, and Xinyu Zhao. "Riemannian Gradient Algorithm for the Numerical Solution of Linear Matrix Equations." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/507175.

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Анотація:

A Riemannian gradient algorithm based on geometric structures of a manifold consisting of all positive definite matrices is proposed to calculate the numerical solution of the linear matrix equationQ=X+∑i=1mAiTXAi. In this algorithm, the geodesic distance on the curved Riemannian manifold is taken as an objective function and the geodesic curve is treated as the convergence path. Also the optimal variable step sizes corresponding to the minimum value of the objective function are provided in order to improve the convergence speed. Furthermore, the convergence speed of the Riemannian gradient algorithm is compared with that of the traditional conjugate gradient method in two simulation examples. It is found that the convergence speed of the provided algorithm is faster than that of the conjugate gradient method.

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40

Suh, Yoon-Je, and Byung Hyung Kim. "Riemannian Embedding Banks for Common Spatial Patterns with EEG-based SPD Neural Networks." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 1 (May 18, 2021): 854–62. http://dx.doi.org/10.1609/aaai.v35i1.16168.

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Анотація:

Modeling non-linear data as symmetric positive definite (SPD) matrices on Riemannian manifolds has attracted much attention for various classification tasks. In the context of deep learning, SPD matrix-based Riemannian networks have been shown to be a promising solution for classifying electroencephalogram (EEG) signals, capturing the Riemannian geometry within their structured 2D feature representation. However, existing approaches usually learn spatial-temporal structures in an embedding space for all available EEG signals, and their optimization procedures rely on computationally expensive iterations. Furthermore, these approaches often struggle to encode all of the various types of relationships into a single distance metric, resulting in a loss of generality. To address the above limitations, we propose a Riemannian Embedding Banks method, which divides the problem of common spatial patterns learning in an entire embedding space into K-subproblems and builds one model for each subproblem, to be combined with SPD neural networks. By leveraging the concept of the "separate to learn" technology on a Riemannian manifold, REB divides the data and the embedding space into K non-overlapping subsets and learns K separate distance metrics in a Riemannian geometric space instead of the vector space. Then, the learned K non-overlapping subsets are grouped into neurons in the SPD neural network's embedding layer. Experimental results on public EEG datasets demonstrate the superiority of the proposed approach for learning common spatial patterns of EEG signals despite their non-stationary nature, increasing the convergence speed while maintaining generalization.

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41

Urakawa, Hajime. "Convergence rates to equilibrium of the heat kernels on compact Riemannian manifolds." Indiana University Mathematics Journal 55, no. 1 (2006): 259–88. http://dx.doi.org/10.1512/iumj.2006.55.2676.

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Fang, Fuquan, and Xiaochun Rong. "The second twisted Betti number and the convergence of collapsing Riemannian manifolds." Inventiones mathematicae 150, no. 1 (October 2002): 61–109. http://dx.doi.org/10.1007/s00222-002-0230-2.

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43

Ferreira, O. P., and R. C. M. Silva. "Local convergence of Newton's method under a majorant condition in Riemannian manifolds." IMA Journal of Numerical Analysis 32, no. 4 (January 30, 2012): 1696–713. http://dx.doi.org/10.1093/imanum/drr048.

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Gao, L. Zhiyong. "Convergence of Riemannian manifolds; Ricci and $L\sp {n/2}$-curvature pinching." Journal of Differential Geometry 32, no. 2 (1990): 349–81. http://dx.doi.org/10.4310/jdg/1214445311.

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45

SCHULTE-HERBRÜGGEN, THOMAS, STEFFEN J. GLASER, GUNTHER DIRR, and UWE HELMKE. "GRADIENT FLOWS FOR OPTIMIZATION IN QUANTUM INFORMATION AND QUANTUM DYNAMICS: FOUNDATIONS AND APPLICATIONS." Reviews in Mathematical Physics 22, no. 06 (July 2010): 597–667. http://dx.doi.org/10.1142/s0129055x10004053.

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Many challenges in quantum information and quantum control root in constrained optimization problems on finite-dimensional quantum systems. The constraints often arise from two facts: (i) quantum dynamic state spaces are naturally smooth manifolds (orbits of the respective initial states) rather than being Hilbert spaces; (ii) the dynamics of the respective quantum system may be restricted to a proper subset of the entire state space. Mathematically, either case can be treated by constrained optimization over the reachable set of an underlying control system. Thus, whenever the reachable set takes the form a smooth manifold, Riemannian optimization methods apply. Here, we give a comprehensive account on the foundations of gradient flows on Riemannian manifolds including new applications in quantum information and quantum dynamics. Yet, we do not pursue the problem of designing explicit controls for the underlying control systems. The framework is sufficiently general for setting up gradient flows on (sub)manifolds, Lie (sub)groups, and (reductive) homogeneous spaces. Relevant convergence conditions are discussed, in particular for gradient flows on compact and analytic manifolds. This is meant to serve as foundation for new achievements and further research. Illustrative examples and new applications are given: we extend former results on unitary groups to closed subgroups with tensor-product structure, where the finest product partitioning relates to SU loc (2n) := SU(2) ⊗ ⋯ ⊗ SU(2) — known as (qubit-wise) local unitary operations. Such applications include, e.g., optimizing figures of merit on SU loc (2n) relating to distance measures of pure-state entanglement as well as to best rank-1 approximations of higher-order tensors. In quantum information, our gradient flows provide a numerically favorable alternative to standard tensor-SVD techniques.

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Liao, Jiagen, and Zhongping Wan. "Inexact Restoration Methods for Semivectorial Bilevel Programming Problem on Riemannian Manifolds." Axioms 11, no. 12 (December 5, 2022): 696. http://dx.doi.org/10.3390/axioms11120696.

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For a better understanding of the bilevel programming on Riemannian manifolds, a semivectorial bilevel programming scheme is proposed in this paper. The semivectorial bilevel programming is firstly transformed into a single-level programming problem by using the Karush–Kuhn–Tucker (KKT) conditions of the lower-level problem, which is convex and satisfies the Slater constraint qualification. Then, the single-level programming is divided into two stages: restoration and minimization, based on which an Inexact Restoration algorithm is developed. Under certain conditions, the stability and convergence of the algorithm are analyzed.

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Zhang, Yachao, Xuan Lai, Yuan Xie, Yanyun Qu, and Cuihua Li. "Geometry-Aware Discriminative Dictionary Learning for PolSAR Image Classification." Remote Sensing 13, no. 6 (March 23, 2021): 1218. http://dx.doi.org/10.3390/rs13061218.

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In this paper, we propose a new discriminative dictionary learning method based on Riemann geometric perception for polarimetric synthetic aperture radar (PolSAR) image classification. We made an optimization model for geometry-aware discrimination dictionary learning in which the dictionary learning (GADDL) is generalized from Euclidian space to Riemannian manifolds, and dictionary atoms are composed of manifold data. An efficient optimization algorithm based on an alternating direction multiplier method was developed to solve the model. Experiments were implemented on three public datasets: Flevoland-1989, San Francisco and Flevoland-1991. The experimental results show that the proposed method learned a discriminative dictionary with accuracies better those of comparative methods. The convergence of the model and the robustness of the initial dictionary were also verified through experiments.

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Habib, Georges, and Ken Richardson. "Riemannian flows and adiabatic limits." International Journal of Mathematics 29, no. 02 (February 2018): 1850011. http://dx.doi.org/10.1142/s0129167x18500118.

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Habermann, Lutz, and Jürgen Jost. "Convergence of eigenvalues and Green functions under surgery type degeneration of Riemannian manifolds." Calculus of Variations and Partial Differential Equations 5, no. 2 (June 1997): 137–58. http://dx.doi.org/10.1007/s005260050063.

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de Carvalho Bento, Glaydston, João Xavier da Cruz Neto, and Paulo Roberto Oliveira. "A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds." Journal of Optimization Theory and Applications 168, no. 3 (January 11, 2016): 743–55. http://dx.doi.org/10.1007/s10957-015-0861-2.

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