Optimization techniques on Riemannian manifolds

Smith, Steven T.

doi:10.1090/fic/003/09

Cited by 229 publications

(311 citation statements)

References 32 publications

(30 reference statements)

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“…This natural Riemannian version of Newton's method for solving systems of nonlinear equations has been considered by several authors [1,5,6,8,19,20]. In order to state a general local convergence result for R-Newton's method, we will assume a Lipschitz-type continuity of X on a neighborhood of p 0 , based on the following definition.…”

Section: Rr N°5381mentioning

confidence: 99%

“…To our best knowledge, the theory of self-concordancy has not been extended to a Riemannian setting yet. The main motivation for enlarging the usual Euclidean setting to Riemannian manifolds stems essentially from the necessity to be able to deal with (equality) nonlinear constraints, especially in minimization problems; see, for instance, the works by Adler et al [1], da Cruz Neto et al [6], Edelman et al [8], Smith [19] and Udriste [20].…”

Section: Introductionmentioning

confidence: 99%

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A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

2006

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Abstract:We consider the problem of finding a singularity of a vector field X on a complete Riemannian manifold. In this regard we prove a unified result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of X. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area.

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Section: Rr N°5381mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

2006

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“…in [62,65,32,21,1] to cite just a few. The iteration we investigate in this paper does not enter into this family as it does not use the covariant derivative of the vector field of which we are trying to find the zeros, Moreover, we cannot recast it as an optimization problem on a Riemannian manifold, as stated above.…”

Section: Related Workmentioning

confidence: 99%

“…Newton algorithms on Riemannian manifolds were first proposed in the general context of optimization on manifolds [62,65]. Their convergence has been studied in depth in [47,32,1] to cite just a few of the important works.…”

Section: A Fixed Point Iteration To Compute the Karcher Meanmentioning

confidence: 99%

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

Pennec

Arsigny

2012

Matrix Information Geometry

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Abstract. When performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this paper, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance.

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“…Algorithms of this form can be found in [LOS00], [Ous99], [MS05]. As explained in [MM05], these methods can be seen as concrete versions of the intrinsic Riemannian Newton method (see [Gab82] and [Smi94] among others).…”

Section: Introductionmentioning

confidence: 99%

Geometrical interpretation of the predictor-corrector type algorithms in structured optimization problems

2006

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It has been observed that in many optimization problems, nonsmooth objective functions often appear smooth on naturally arising manifolds. This has led to the development of optimization algorithms which attempt to exploit this smoothness. Many of these algorithms follow the same two step pattern: first predict a direction of decrease, and second make a correction step to return to the manifold. In this paper we examine some of the theoretical components used in such predictor-corrector methods. We begin our examination under the minimal assumption that the restriction of the function to the manifold is smooth. At the second stage, we add the condition of "partial smoothness" relative to the manifold. Finally, we examine the case when the function is both "prox-regular" and partly smooth. In this final setting we show that the proximal point mapping can be used to return to the manifold, and argue that returning in this manner is preferable to returning via the projection mapping. We finish by developing sufficient conditions for quadratic convergence of predictor-corrector methods using a proximal point correction step.

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Optimization techniques on Riemannian manifolds

Cited by 229 publications

References 32 publications

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

Geometrical interpretation of the predictor-corrector type algorithms in structured optimization problems

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