#### DMCA

## dans le cadre de l ’ École Doctorale Science de l’Ingénieur et l’Information (2012)

### Citations

219 | Optimal transport, old and new - Villani - 2008 |

201 | Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements
- Pennec
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Citation Context ... of Karcher means. Gradient descent methods for computing Karcher means of probability measures in Riemannian manifolds are proposed by many authors. For example, by H. Le in [63] and by X. Pennec in =-=[74]-=-. New stochastic and deterministic algorithms with explicit stepsizes and error estimates for computing the Fréchet p-means of probability measures are given in Chapter 4 of this dissertation. Medians... |

124 | Incremental subgradient methods for nondifferentiable optimization
- Nedic, Bertsekas
(Show Context)
Citation Context ...ns of the stepsizes. In Euclidean spaces, it is well known that the following type of inequalities are of fundamental importance to conclude the convergence of subgradient algorithms (see for example =-=[34, 71]-=-): ||xk+1 − y|| 2 ≤ ||xk − y|| 2 + αt 2 k 2tk + β (f(y) − f(xk)). ||vk|| For a nonnegatively curved Riemannian manifold, Ferreira and Oliveira obtained a generalization of the above inequality in [40]... |

116 |
Sur le point pour lequel la somme des distances de n points donnes est minimum.
- Weiszfeld
- 1937
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Citation Context ...eralized Fermat’s problem cannot be solved by arithmetic operations and extraction of roots. The first algorithm intended to find medians in Euclidean spaces was proposed by E. Weiszfeld in 1937 (see =-=[90]-=-), whose original idea is to construct a sequence of weighted barycenters of data points which converges to the median. But this algorithm, which is essentially a gradient descent method, has a2 CHAP... |

114 |
The Schur Complement and Its Applications
- Zhang
- 2005
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Citation Context ...1 : . 0 ⎥ ⎦ , ϕ : Tn −→ R ∗ + × Dn−1 , Rn ↦−→ (P0,µ1,...,µn−1), where D = {z ∈ C : |z| < 1} is the unit disc of the complex plane. Using the Cramer’s rule and the method of Schur complement (see e.g. =-=[95]-=-) we get the following proposition. Proposition 1.31. ϕ is a diffeomorphism, whose explicit expression is ( ) k det Sk 2,... ,k + 1 µk = (−1) , where Sk = Rk+1 detRk 1,... ,k is the submatrix of Rk+1 ... |

106 | Optimization methods on Riemannian manifolds - Udri¸ste - 1995 |

64 | Convergence rate of incremental subgradient algorithms
- Nedić, Bertsekas
- 2000
(Show Context)
Citation Context ... are obtained by induction. We proceed to show that if (tk)k is chosen to be the harmonic series, then the rate of convergence of our algorithm is sublinear. To do this, we use the following lemma in =-=[70]-=-. Lemma 2.28. Let (uk)k≥0 be a sequence of nonnegative real numbers such that ( uk+1 ≤ 1 − α ) ζ uk + , k + 1 (k + 1) 2 where α and ζ are positive constants. Then ⎧ 1 (k + 2) α ( u0 + 2α ) ζ(2 − α) , ... |

46 | Riemannian Geometry. Translations of Mathematical Monographs 149 - Sakai - 1996 |

37 |
Multidimensional diffusion processes, Grundlehren der
- Stroock, Varadhan
- 1979
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Citation Context ...n) ε (ε) n≥1 n≥1 that weakly converges in TeM to the distribution νε. Thanks to Skorohod theorem which allows to realize it as an a.s. convergence and to lemma 4.11 we ( can apply ) Theorem 11.2.3 of =-=[82]-=-, and we obtain that the sequence of processes ˜Y ψ(n) ε n≥1 weakly converges to a diffusion (yt)ε≤t≤T with generator Gδ(t) given by (3.4) and such that yε has law νε. This achieves the proof of lemma... |

27 |
On the convergence of a class of iterative methods for solving the Weber location problem
- Ostresh
- 1978
(Show Context)
Citation Context ...he median coincides with some data point, which is clearly an event of positive probability. The first completely convergent algorithm for computing medians was proposed by L. M. Ostresh in 1978 (see =-=[73]-=-), whose key ingredient is to eliminate the singular term when the algorithm hits a data point. Since then various algorithms for computing medians in Euclidean spaces were proposed and improved by ma... |

19 |
The geometry of Hessian structures; World Scientific Publishing Co. Pte
- Shima
- 2007
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Citation Context ...ons of particular forms. A similar potential function was used by S. Amari in [5] to derive the Riemannian metric of multi-variate Gaussian distributions by means of divergence functions. We refer to =-=[78]-=- for more account on the geometry of Hessian structures. With the metric given by (2.10) the space R ∗ + ×Dn−1 is just the product of the Riemannian manifolds (R ∗ + ,ds2 0 ) and (D,ds2 k )1≤k≤n−1, wh... |

17 |
A modified Weiszfeld algorithm for the Fermat-Weber location problem,
- Vardi, Zhang
- 2001
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Citation Context ...e medians. The idea to add the penalty term µ{x}d(y,x) in the definition of hx is to ensure that the fixed point mapping T diminishes the value of f. The main results of Chapter 5 generalize those of =-=[87]-=-, in which all the results are proved only in Euclidean spaces. The framework of Chapter 5 is the same to that of Chapter 2, so I omit it here. Besides, we shall make some assumption on the probabilit... |

13 |
Riemannian Median and Its Estimation. In:
- Yang
- 2010
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Citation Context ...let t ↦→ E(t) := 1 2 ρ2 (e,γ(t)) , γ(t) t∈[0,tk+1] the geodesic satisfying ˙γ(0) = − grad Xk Fp(·,Pk+1). We have for all t ∈ [0,tk+1] E ′′ (t) ≤ C(β,r,p) := p 2 (2r) 2p−1 β coth(2βr) (2.12) (see e.g. =-=[91]-=-). By Taylor formula, ρ(Xk+1,e) 2 = 2E(tk+1) = 2E(0) + 2tk+1E ′ (0) + t 2 k+1 E′′ (t) for some t ∈ [0,tk+1] ≤ ρ(Xk,e) 2 + 2tk+1〈grad Xk Fp(·,Pk+1),exp −1 Xk (e)〉 + t2 k+1 C(β,r,p). Now from the convex... |

12 |
Barycentres et martingales sur une variété,
- Picard
- 1994
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Citation Context ...shed by stochastic flows is also considered in [9]. In order to study harmonic maps between Riemannian manifolds with probabilistic methods, J. Picard also gave a generalized notion of barycenters in =-=[75]-=-. As we noted before, Karcher means are only local minimizers of the energy functional fµ in (1.2), but it is easily seen that fµ can be defined not only on the closed ball ¯ B(a,ρ) but also on the wh... |

8 |
Uber den Standort der Industrien, Teil 1: Reine Theorie des Standorts,
- Weber
- 1909
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Citation Context ...econd author showed that the minimum point must be unique if the data points are not contained in a single line. This generalized Fermat’s problem was also used by the economist A. Weber in 1909 (see =-=[89]-=-) as a mathematical model for the optimal location of a facility in order to serve several clients. From then on, the generalized Fermat’s problem is also called Steiner’s problem or Weber’s problem o... |

7 | Espérance d’une variable aléatoire á valeurs dans un espace métrique, Thése de l’université de Rouen - Sahib - 1998 |

4 | Some properties of Frechet medians in Riemannian manifolds, preprint 22 M. ARNAUDON AND L. MICLO Laboratoire de Mathématiques et Applications CNRS - Yang |

3 |
Generalized Fermat’s problem
- Noda, Sakai, et al.
- 1991
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Citation Context ...y out robust estimations for directional data. Perhaps the first work about Fréchet medians on Riemannian manifolds is the paper named “Generalized Fermat’s problem” by R. Noda and his coauthors (see =-=[72]-=-). They proved the uniqueness, characterizations and position estimations of Fréchet medians for discrete sample points lying in a Cartan-Hadamard manifold. By the way, it seems that, probably due to ... |

2 |
1838) Von den Krümmungs-Schwerpuncte ebenen Curven, Journal fûr die reine und angewandte Mathematik
- Steiner
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Citation Context ...an can be generalized to finding a point that minimizes the sum of its distances to N given points in the plan and, if necessary, with weighted distances. This was proposed by J. Steiner in 1838 (see =-=[81]-=-) and was also considered by R. Sturm in 1884 (see [83]). The second author showed that the minimum point must be unique if the data points are not contained in a single line. This generalized Fermat’... |

1 | Mathématique Sociale” and Mathematics. A case study: Condorcet’s effect and medians, Electronic Journ@l for History of Probability and Statistics, vol 4, n ◦ 1; juin/june 2008 - Monjardet |

1 | some properties of positive definite Toeplitz Matrices and their possible applications, Linear algebra and its applications - Mukherjee, MaitiOn - 1988 |

1 | SkovgaardA riemannian geometry of the multivariate normal model - T - 1984 |

1 | Multidimensional medians arising from geodesics on graphs - Small - 1997 |

1 |
Ueber den Punkt Keinster Entfernugssumme von gegebenen Punkten, Journal fûr die reine und angewante Mathematik, vol 97
- Sturm
(Show Context)
Citation Context ... the sum of its distances to N given points in the plan and, if necessary, with weighted distances. This was proposed by J. Steiner in 1838 (see [81]) and was also considered by R. Sturm in 1884 (see =-=[83]-=-). The second author showed that the minimum point must be unique if the data points are not contained in a single line. This generalized Fermat’s problem was also used by the economist A. Weber in 19... |

1 |
Vaidyanathan The Theory of Linear Prediction, Morgan and Claypool
- P
- 2012
(Show Context)
Citation Context ... k is called the k-th reflection coefficient and is denoted by µk. It is easily seen that µ1,... ,µn−1 are uniquely determined by the matrix Rn. Moreover, the classical Levinson’s recursion (see e.g. =-=[86]-=-) gives that |µk| < 1. Hence, by letting P0 = r0, we obtain a map between two submanifolds of R2n−1 : . 0 ⎥ ⎦ , ϕ : Tn −→ R ∗ + × Dn−1 , Rn ↦−→ (P0,µ1,...,µn−1), where D = {z ∈ C : |z| < 1} is the uni... |

1 | Riemannian median, geometry of covariance matrices and radar target detection, European Radar Conference 2010 - Yang, Arnaudon, et al. |

1 | Geometry of covariance matrices and computation - Yang, Arnaudon, et al. |