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## Optimization techniques on Riemannian manifolds (1994)

Venue: | FIELDS INSTITUTE COMMUNICATIONS |

Citations: | 84 - 1 self |

### Citations

2704 |
Matrix Computations
- Golub, Loan
- 1996
(Show Context)
Citation Context ...tructure of the manifold. Elements of this spirit may be found throughout the field of numerical methods, such as the emphasis on unitary (norm preserving) transformations in numerical linear algebra =-=[22]-=-, or the use of feasible direction methods [18, 21, 38]. An intrinsic approach leads one from the extrinsic idea of vector addition to the exponential map and parallel translation, from minimization a... |

1778 |
Differential Geometry, Lie Groups, and Symmetric Spaces
- Helgason
- 1978
(Show Context)
Citation Context ...ity of these computations. For example, on a real compact semisimple Lie group endowed with its natural Riemannian metric, geodesics and parallel translation may be computed via matrix exponentiation =-=[24]-=-. Several algorithms are available to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovic... |

1475 |
Introduction to Linear and Nonlinear Programming
- Luenberger
- 1973
(Show Context)
Citation Context ...f steepest descent on Euclidean space. Each iteration involves a gradient computation and minimization along the geodesic determined by the gradient. Fletcher [18], Botsaris [5, 6, 7], and Luenberger =-=[31]-=- describe this algorithm in Euclidean space. Gill and Murray [21] and Sargent [38] apply this technique in the presence of constraints. In this section we restate the method of steepest descent descri... |

1132 |
A Comprehensive Introduction to Differential Geometry
- Spivak
- 1979
(Show Context)
Citation Context ...an notions of straight lines and ordinary differentiation with geodesics and covariant differentiation. These concepts are reviewed in the following paragraphs. We follow Helgason’s [24] and Spivak’s =-=[46]-=- treatments of covariant differentiation, the exponential map, and parallel translation. Details may be found in these references. Let M be a complete n-dimensional Riemannian manifold with Riemannian... |

1093 |
Methods of conjugate gradients for solving linear systems
- Hestenes, Stiefel
- 1952
(Show Context)
Citation Context ...ant differential, and its inverse. In this section we describe the conjugate gradient method, which has the dual advantages of algorithmic simplicity and superlinear convergence. Hestenes and Stiefel =-=[26]-=- first used conjugate gradient methods to compute the solutions of linear equations, or, equivalently, to compute the minimum of a quadratic form on Rn. This approach can be modified to yield effectiv... |

975 | The symmetric Eigenvalue Problem - Parlett - 1980 |

421 | Nineteen dubious ways to compute the exponential of a matrix, twenty-fiveyearslater
- Moler, Loan
- 1994
(Show Context)
Citation Context ...e Lie group endowed with its natural Riemannian metric, geodesics and parallel translation may be computed via matrix exponentiation [24]. Several algorithms are available to perform this computation =-=[22, 32]-=-. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias [30], Chu et al. [13, 14], Perkins et al. [35], and He... |

372 |
Function minimization by conjugate gradients
- Fletcher, Reeves
- 1964
(Show Context)
Citation Context ...to compute the minimum of a quadratic form on Rn. This approach can be modified to yield effective algorithms to compute the minima of nonquadratic functions on Rn. In particular, Fletcher and Reeves =-=[19]-=- and Polak and Ribière [36] provide algorithms based upon the assumption that the second order Taylor expansion of the function to be minimized sufficiently approximates this function near the minimu... |

256 |
Riemannian center of mass and mollifier smoothing
- Karcher
- 1977
(Show Context)
Citation Context ...es). Nevertheless, the amount by which exp fails to preserve inner products can be quantified via the Gauss Lemma and Jacobi’s equation; see, e.g., Cheeger and Ebin [12], or the appendices of Karcher =-=[28]-=-. Let t be small, and let X ∈ Tp̂ and Y ∈ TtX(Tp̂) ∼= Tp̂ be orthonormal tangent vectors. The amount by which the exponential map changes the length of tangent vectors is approximated by the Taylor ex... |

251 | Stable Mappings and Their Singularities - Golubitsky, Guillemin - 1973 |

250 |
Comparison theorems in Riemannian geometry
- Cheeger, Ebin
- 1975
(Show Context)
Citation Context ... of 0 ∈ Rn via the normal coordinates). Nevertheless, the amount by which exp fails to preserve inner products can be quantified via the Gauss Lemma and Jacobi’s equation; see, e.g., Cheeger and Ebin =-=[12]-=-, or the appendices of Karcher [28]. Let t be small, and let X ∈ Tp̂ and Y ∈ TtX(Tp̂) ∼= Tp̂ be orthonormal tangent vectors. The amount by which the exponential map changes the length of tangent vecto... |

142 | Projected Newton methods for optimization problems with simple constraints
- Bertsekas
- 1982
(Show Context)
Citation Context ... method will rely upon the Taylor expansion of the one-form df . Note that Newton’s method has a counterpart in the theory of constrained optimization, as described by, e.g., Fletcher [18], Bertsekas =-=[1, 2]-=-, or Dunn [15, 16]. The Newton method presented in this section has only local convergence properties. There is a theory of global Newton methods on Euclidean space and computational complexity; see t... |

139 |
Computational Methods in Optimization
- Polak
- 1971
(Show Context)
Citation Context ...quadratic form on Rn. This approach can be modified to yield effective algorithms to compute the minima of nonquadratic functions on Rn. In particular, Fletcher and Reeves [19] and Polak and Ribière =-=[36]-=- provide algorithms based upon the assumption that the second order Taylor expansion of the function to be minimized sufficiently approximates this function near the minimum. In addition, Davidon, Fle... |

82 |
Invariant affine connections on homogeneous spaces
- Nomizu
- 1954
(Show Context)
Citation Context ... 14], Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting parallel translation, a reductive homogeneous space =-=[29, 33]-=-. Perhaps the simplest nontrivial example is the sphere, where geodesics and parallel translation can be computed at low cost with trigonometric functions and vector addition. Furthermore, Brown and B... |

59 | The projected gradient method for least squares matrix approximation with spectral constraints
- Chu, Driessel
(Show Context)
Citation Context ... to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias [30], Chu et al. =-=[13, 14]-=-, Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting parallel translation, a reductive homogeneous space [29,... |

34 |
Some remarks on dynamical systems and numerical analysis, in Dynamical systems and partial differential equations
- Shub
- 1984
(Show Context)
Citation Context ...der as—the Rayleigh quotient iteration; thus, the RQI is seen to be an efficient approximation of Newton’s method. If the exponential map is replaced by the chart v ∈ Tx 7→ (x+ v)/‖x+ v‖ ∈ Sn−1, Shub =-=[39]-=- shows that a corresponding version of Newton’s method is equivalent to the RQI. Example 4.10 (The function tr ΘTQΘN). Let Θ, Q, H = AdΘT(Q), and Ω be as in Example 3.6. The second covariant different... |

28 |
Least square matching problems
- Brockett
- 1989
(Show Context)
Citation Context ...nslation may be computed via matrix exponentiation [24]. Several algorithms are available to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett =-=[8, 9, 10]-=-, Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias [30], Chu et al. [13, 14], Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a... |

27 |
Computational complexity: on the geometry of polynomials and a theory of cost II
- Shub, Smale
- 1986
(Show Context)
Citation Context ...nly local convergence properties. There is a theory of global Newton methods on Euclidean space and computational complexity; see the work of Hirsch and Smale [27], Smale [43, 44], and Shub and Smale =-=[40, 41]-=-. Let M be an n-dimensional Riemannian manifold with Riemannian structure g and Levi-Civita connection ∇, let µ be a C∞ one-form on M , and let p in M be such that the bilinear form (∇µ)p:Tp × Tp → R ... |

18 |
A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map
- Bloch, Brockett, et al.
- 1990
(Show Context)
Citation Context ...via matrix exponentiation [24]. Several algorithms are available to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. =-=[3, 4]-=-, Smith [45], Faybusovich [17], Lagarias [30], Chu et al. [13, 14], Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, ... |

18 |
On algorithms for solving f(x
- Hirsch
(Show Context)
Citation Context ...on method presented in this section has only local convergence properties. There is a theory of global Newton methods on Euclidean space and computational complexity; see the work of Hirsch and Smale =-=[27]-=-, Smale [43, 44], and Shub and Smale [40, 41]. Let M be an n-dimensional Riemannian manifold with Riemannian structure g and Levi-Civita connection ∇, let µ be a C∞ one-form on M , and let p in M be s... |

15 |
Hamiltonian structure of dynamical systems which solve linear programming problems
- FAYBUSOVICH
- 1991
(Show Context)
Citation Context ...Several algorithms are available to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovich =-=[17]-=-, Lagarias [30], Chu et al. [13, 14], Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting parallel translation... |

14 |
Some effective methods for unconstrained optimization based on the solution of systems of ordinary differential equations
- Brown, Bartholomew-Biggs
- 1989
(Show Context)
Citation Context ...plest nontrivial example is the sphere, where geodesics and parallel translation can be computed at low cost with trigonometric functions and vector addition. Furthermore, Brown and Bartholomew-Biggs =-=[11]-=- show that in some cases function minimization by following the solution of a system of ordinary differential equations can be implemented such that it is competitive with conventional techniques. The... |

12 |
Balanced realizations via gradient flow techniques. Systems and Control
- Perkins, Helmke, et al.
- 1990
(Show Context)
Citation Context ...ion [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias [30], Chu et al. [13, 14], Perkins et al. =-=[35]-=-, and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting parallel translation, a reductive homogeneous space [29, 33]. Perhaps the sim... |

9 |
Newton’s Method and the Goldstein Step-Length Rule for Constrained Minimization Problems
- Dunn
- 1980
(Show Context)
Citation Context ...y upon the Taylor expansion of the one-form df . Note that Newton’s method has a counterpart in the theory of constrained optimization, as described by, e.g., Fletcher [18], Bertsekas [1, 2], or Dunn =-=[15, 16]-=-. The Newton method presented in this section has only local convergence properties. There is a theory of global Newton methods on Euclidean space and computational complexity; see the work of Hirsch ... |

8 |
An iterative algorithm for locating the minimal eigenvector of a symmetric matrix
- Fuhrmann, Liu
- 1984
(Show Context)
Citation Context ... minimization or computation of ρ(hi), and 10n flops per iteration. The results of a numerical experiment demonstrating the convergence of Algorithm 5.5 on S20 are shown in Figure 1. Fuhrmann and Liu =-=[20]-=- provide a conjugate gradient algorithm for Rayleigh’s quotient on the sphere that uses an azimuthal projection onto tangent planes. Example 5.6 (The function tr ΘTQΘN). Let Θ, Q, and H be as in Examp... |

8 |
Dynamical systems that perform the singular value decomposition
- Smith
- 1991
(Show Context)
Citation Context ...onentiation [24]. Several algorithms are available to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith =-=[45]-=-, Faybusovich [17], Lagarias [30], Chu et al. [13, 14], Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting pa... |

7 |
Quasi-Newton methods for linearly constrained optimization
- Gill, Murray
- 1973
(Show Context)
Citation Context ...it may be found throughout the field of numerical methods, such as the emphasis on unitary (norm preserving) transformations in numerical linear algebra [22], or the use of feasible direction methods =-=[18, 21, 38]-=-. An intrinsic approach leads one from the extrinsic idea of vector addition to the exponential map and parallel translation, from minimization along lines to minimization along geodesics, and from pa... |

6 |
Differential gradient methods
- Botsaris
- 1978
(Show Context)
Citation Context ... identical to the method of steepest descent on Euclidean space. Each iteration involves a gradient computation and minimization along the geodesic determined by the gradient. Fletcher [18], Botsaris =-=[5, 6, 7]-=-, and Luenberger [31] describe this algorithm in Euclidean space. Gill and Murray [21] and Sargent [38] apply this technique in the presence of constraints. In this section we restate the method of st... |

6 |
Eigensystem computation for skew-symmetric matrices and a class of symmetric matrices
- Ward, Gray
- 1978
(Show Context)
Citation Context ...matrix of eigenvalues of Hi, H0 is near N , and ‖ · ‖ is the norm induced by the standard inner product on gl(n). Geodesics and parallel translation were computed using the algorithm of Ward and Gray =-=[47,48]-=-; the step sizes for the method of steepest descent and the conjugate gradient method were computed using Brockett’s estimate [10]. by g. Brockett’s estimate (n.b. Eq. (13)) for the step size may be u... |

5 |
Isospectral flows on symmetric matrices and the Riccati equation
- Helmke
- 1991
(Show Context)
Citation Context ...s algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias [30], Chu et al. [13, 14], Perkins et al. [35], and Helmke =-=[25]-=-. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting parallel translation, a reductive homogeneous space [29, 33]. Perhaps the simplest nontrivial ... |

2 |
systems that sort lists, diagonalize matrices, and solve linear programming problems
- Dynamical
- 1991
(Show Context)
Citation Context ...nslation may be computed via matrix exponentiation [24]. Several algorithms are available to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett =-=[8, 9, 10]-=-, Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias [30], Chu et al. [13, 14], Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a... |

2 |
and asymptotic convergence rate estimates for a class of projected gradient processes
- Global
- 1981
(Show Context)
Citation Context ...y upon the Taylor expansion of the one-form df . Note that Newton’s method has a counterpart in the theory of constrained optimization, as described by, e.g., Fletcher [18], Bertsekas [1, 2], or Dunn =-=[15, 16]-=-. The Newton method presented in this section has only local convergence properties. There is a theory of global Newton methods on Euclidean space and computational complexity; see the work of Hirsch ... |

2 |
Practical Methods of Optimization. 2d ed
- Fletcher
- 1987
(Show Context)
Citation Context ...it may be found throughout the field of numerical methods, such as the emphasis on unitary (norm preserving) transformations in numerical linear algebra [22], or the use of feasible direction methods =-=[18, 21, 38]-=-. An intrinsic approach leads one from the extrinsic idea of vector addition to the exponential map and parallel translation, from minimization along lines to minimization along geodesics, and from pa... |

2 |
Monotonicity properties of the Toda flow, the QR-flow, and subspace iteration
- Lagarias
- 1991
(Show Context)
Citation Context ...hms are available to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias =-=[30]-=-, Chu et al. [13, 14], Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting parallel translation, a reductive h... |

2 |
Reduced gradient and projection methods for nonlinear programming, in Numerical Methods for Constrained Optimization
- Sargent
- 1974
(Show Context)
Citation Context ...it may be found throughout the field of numerical methods, such as the emphasis on unitary (norm preserving) transformations in numerical linear algebra [22], or the use of feasible direction methods =-=[18, 21, 38]-=-. An intrinsic approach leads one from the extrinsic idea of vector addition to the exponential map and parallel translation, from minimization along lines to minimization along geodesics, and from pa... |

2 |
complexity: On the geometry of polynomials and a theory of cost
- Computational
- 1986
(Show Context)
Citation Context ...nly local convergence properties. There is a theory of global Newton methods on Euclidean space and computational complexity; see the work of Hirsch and Smale [27], Smale [43, 44], and Shub and Smale =-=[40, 41]-=-. Let M be an n-dimensional Riemannian manifold with Riemannian structure g and Levi-Civita connection ∇, let µ be a C∞ one-form on M , and let p in M be such that the bilinear form (∇µ)p:Tp × Tp → R ... |

2 |
The fundamental theorem of algebra and computational complexity
- Smale
- 1981
(Show Context)
Citation Context ...esented in this section has only local convergence properties. There is a theory of global Newton methods on Euclidean space and computational complexity; see the work of Hirsch and Smale [27], Smale =-=[43, 44]-=-, and Shub and Smale [40, 41]. Let M be an n-dimensional Riemannian manifold with Riemannian structure g and Levi-Civita connection ∇, let µ be a C∞ one-form on M , and let p in M be such that the bil... |

2 |
Algorithm 530: An Algorithm for Computing the Eigensystem of Skew-Symmetric Matrices and a Class of Symmetric Matrices
- Ward, Gray
- 2005
(Show Context)
Citation Context ...matrix of eigenvalues of Hi, H0 is near N , and ‖ · ‖ is the norm induced by the standard inner product on gl(n). Geodesics and parallel translation were computed using the algorithm of Ward and Gray =-=[47,48]-=-; the step sizes for the method of steepest descent and the conjugate gradient method were computed using Brockett’s estimate [10]. by g. Brockett’s estimate (n.b. Eq. (13)) for the step size may be u... |

1 |
optimization along geodesics
- Constrained
- 1981
(Show Context)
Citation Context ... identical to the method of steepest descent on Euclidean space. Each iteration involves a gradient computation and minimization along the geodesic determined by the gradient. Fletcher [18], Botsaris =-=[5, 6, 7]-=-, and Luenberger [31] describe this algorithm in Euclidean space. Gill and Murray [21] and Sargent [38] apply this technique in the presence of constraints. In this section we restate the method of st... |

1 |
Curves on Sn−1 that lead to eigenvalues or their means of a matrix
- Chu
- 1986
(Show Context)
Citation Context ... to perform this computation [22, 32]. This algebraic structure may be found in the problems posed by Brockett [8, 9, 10], Bloch et al. [3, 4], Smith [45], Faybusovich [17], Lagarias [30], Chu et al. =-=[13, 14]-=-, Perkins et al. [35], and Helmke [25]. This approach is also applicable if the manifold can be identified with a symmetric space or, excepting parallel translation, a reductive homogeneous space [29,... |