Convex Functions on Riemannian Manifolds

Udrişte, Constantin

doi:10.1007/978-94-015-8390-9_3

Cited by 43 publications

(45 citation statements)

References 4 publications

(6 reference statements)

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“…A number of authors have proposed and developed a general theory of Newton iteration on Riemannian manifolds [16,28,32,33,35]. In particular, Smith [33] proposes an algorithm for abstract Riemannian manifolds which amounts to the following.…”

Section: Newton Iteration On the Grassmann Manifoldmentioning

confidence: 99%

“…Equation (34) comes from the fact that the metric tensor g ij and the derivatives of φ are smooth functions defined on a compact set, thus bounded. Equation (35) comes because g ij is nondegenerate and smooth on a compact set.…”

Section: Appendix a Quadratic Convergence Of Riemann-newtonmentioning

confidence: 99%

“…Yet many computational problems are posed on non-Euclidean spaces. Several authors [16,27,28,33,35] have proposed abstract algorithms that exploit the underlying geometry (e.g., symmetric, homogeneous, Riemannian) of manifolds on which problems are cast, but the conversion of these abstract geometric algorithms into numerical procedures in practical situations is often a nontrivial task that critically relies on an adequate representation of the manifold.…”

Section: Introductionmentioning

confidence: 99%

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Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

Absil¹,

Mahony²,

Sepulchre³

2004

Acta Applicandae Mathematicae

311

389

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Abstract. We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R n . In these formulas, p-planes are represented as the column space of n × p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications -computing an invariant subspace of a matrix and the mean of subspaces -are worked out. (2000): 65J05, 53C05, 14M15. Mathematics Subject Classifications

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Section: Newton Iteration On the Grassmann Manifoldmentioning

confidence: 99%

Section: Appendix a Quadratic Convergence Of Riemann-newtonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Riemannian Geometry of Grassmann Manifolds with a View on Algorithmic Computation

Absil¹,

Mahony²,

Sepulchre³

2004

Acta Applicandae Mathematicae

311

389

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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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