Edit on GitHub

Literature

This is all literature mentioned / referenced in the Manopt.jl documenation. Usually you will find a small reference section at the end of every documentation page that contains references.

[ABG06]
P.-A. Absil, C. Baker and K. Gallivan. Trust-Region Methods on Riemannian Manifolds. Foundations of Computational Mathematics 7, 303–330 (2006).
[AMS08]
P.-A. Absil, R. Mahony and R. Sepulchre. Optimization Algorithms on Matrix Manifolds. Princeton University Press (2008). [open access](http://press.princeton.edu/chapters/absil/).
[AOT22]
S. Adachi, T. Okuno and A. Takeda. Riemannian Levenberg-Marquardt Method with Global and Local Convergence Properties. ArXiv Preprint (2022).
[ABBC20]
N. Agarwal, N. Boumal, B. Bullins and C. Cartis. Adaptive regularization with cubics on manifolds. Mathematical Programming (2020).
[ACOO20]
Y. T. Almeida, J. X. Cruz Neto, P. R. Oliveira and J. C. Oliveira Souza. A modified proximal point method for DC functions on Hadamard manifolds. Computational Optimization and Applications 76, 649–673 (2020).
[Bac14]
M. Bačák. Computing medians and means in Hadamard spaces. SIAM Journal on Optimization 24, 1542–1566 (2014), arXiv: [1210.2145](https://arxiv.org/abs/1210.2145).
[BBSW16]
M. Bačák, R. Bergmann, G. Steidl and A. Weinmann. A second order non-smooth variational model for restoring manifold-valued images. SIAM Journal on Scientific Computing 38, A567–A597 (2016), arxiv: [1506.02409](https://arxiv.org/abs/1506.02409).
[Bea72]
E. M. Beale. A derivation of conjugate gradients. In: Numerical methods for nonlinear optimization, 39–43, London, Academic Press, London (1972).
[BFSS23]
R. Bergmann, O. P. Ferreira, E. M. Santos and J. C. Souza. The difference of convex algorithm on Hadamard manifolds, arXiv preprint (2023).
[BG18]
R. Bergmann and P.-Y. Gousenbourger. A variational model for data fitting on manifolds by minimizing the acceleration of a Bézier curve. Frontiers in Applied Mathematics and Statistics 4 (2018), arXiv: [1807.10090](https://arxiv.org/abs/1807.10090).
[BH19]
R. Bergmann and R. Herzog. Intrinsic formulation of KKT conditions and constraint qualifications on smooth manifolds. SIAM Journal on Optimization 29, 2423–2444 (2019), arXiv: [1804.06214](https://arxiv.org/abs/1804.06214).
[BHS+21]
R. Bergmann, R. Herzog, M. Silva Louzeiro, D. Tenbrinck and J. Vidal-Núñez. Fenchel duality theory and a primal-dual algorithm on Riemannian manifolds. Foundations of Computational Mathematics 21, 1465–1504 (2021), arXiv: [1908.02022](http://arxiv.org/abs/1908.02022).
[BLSW14]
R. Bergmann, F. Laus, G. Steidl and A. Weinmann. Second order differences of cyclic data and applications in variational denoising. SIAM Journal on Imaging Sciences 7, 2916–2953 (2014), arxiv: [1405.5349](https://arxiv.org/abs/1405.5349).
[BPS16]
R. Bergmann, J. Persch and G. Steidl. A parallel Douglas Rachford algorithm for minimizing ROF-like functionals on images with values in symmetric Hadamard manifolds. SIAM Journal on Imaging Sciences 9, 901–937 (2016), arxiv: [1512.02814](https://arxiv.org/abs/1512.02814).
[BIA10]
P. B. Borckmans, M. Ishteva and P.-A. Absil. A Modified Particle Swarm Optimization Algorithm for the Best Low Multilinear Rank Approximation of Higher-Order Tensors. In: 7th International Conference on Swarm INtelligence, editors, 13–23. Springer Berlin Heidelberg (2010).
[Bou23]
N. Boumal. An Introduction to Optimization on Smooth Manifolds. Cambridge University Press (2023).
[Car92]
M. P. do Carmo. Riemannian Geometry. Birkhäuser Boston, Inc., Boston, MA (1992).
[Cas59]
P. de Casteljau. Outillage methodes calcul. Enveloppe Soleau 40.040, Institute National de la Propriété Industrielle, Paris. (1959).
[Cas63]
P. de Casteljau. Courbes et surfaces à pôles. Microfiche P 4147-1, Institute National de la Propriété Industrielle, Paris. (1963).
[CP11]
A. Chambolle and T. Pock. A first-order primal-dual algorithm for convex problems with applications to imaging. Journal of Mathematical Imaging and Vision 40, 120–145 (2011).
[CGT00]
A. R. Conn, N. I. Gould and P. L. Toint. Trust Region Methods. Society for Industrial and Applied Mathematics (2000).
[DY99]
Y. H. Dai and Y. Yuan. A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property. SIAM Journal on Optimization 10, 177–182 (1999).
[DL21]
W. Diepeveen and J. Lellmann. An Inexact Semismooth Newton Method on Riemannian Manifolds with Application to Duality-Based Total Variation Denoising. SIAM Journal on Imaging Sciences 14, 1565–1600 (2021), arXiv: [2102.10309](https://arxiv.org/abs/2102.10309).
[DMSC16]
J. Duran, M. Moeller, C. Sbert and D. Cremers. Collaborative Total Variation: A General Framework for Vectorial TV Models. SIAM Journal on Imaging Sciences 9, 116-151 (2016), arxiv: [1508.01308](https://arxiv.org/abs/1508.01308).
[Fle13]
P. T. Fletcher. Geodesic regression and the theory of least squares on Riemannian manifolds. International Journal of Computer Vision 105, 171–185 (2013).
[Fle87]
R. Fletcher. Practical Methods of Optimization. John Wiley & Sons Ltd. (1987).
[FR64]
R. Fletcher and C. M. Reeves. Function minimization by conjugate gradients. The Computer Journal 7, 149–154 (1964).
[GS23]
G. N. Grapiglia and G. F. Stella. An Adaptive Riemannian Gradient Method Without Function Evaluations. Journal of Optimization Theory and Applications 197, 1140–1160 (2023), preprint: [optimization-online.org/wp-content/uploads/2022/04/8864.pdf](https://optimization-online.org/wp-content/uploads/2022/04/8864.pdf).
[HZ06]
W. W. Hager and H. Zhang. A survey of nonlinear conjugate gradient methods. Pacific Journal of Optimization 2, 35–58 (2006).
[HZ05]
W. W. Hager and H. Zhang. A New Conjugate Gradient Method with Guaranteed Descent and an Efficient Line Search. SIAM Journal on Optimization 16, 170–192 (2005).
[HS52]
M. Hestenes and E. Stiefel. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 49, 409 (1952).
[Hua14]
W. Huang. Optimization algorithms on Riemannian manifolds with applications. Phd thesis, Flordia State University (2014).
[HAG18]
W. Huang, P.-A. Absil and K. A. Gallivan. A Riemannian BFGS method without differentiated retraction for nonconvex optimization problems. SIAM Journal on Optimization 28, 470–495 (2018).
[HGA15]
W. Huang, K. A. Gallivan and P.-A. Absil. A Broyden class of quasi-Newton methods for Riemannian optimization. SIAM Journal on Optimization 25, 1660–1685 (2015).
[IP17]
B. Iannazzo and M. Porcelli. The Riemannian Barzilai{\textendash}Borwein method with nonmonotone line search and the matrix geometric mean computation. IMA Journal of Numerical Analysis 38, 495–517 (2017).
[Kar77]
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30, 509–541 (1977).
[LNPS17]
F. Laus, M. Nikolova, J. Persch and G. Steidl. A nonlocal denoising algorithm for manifold-valued images using second order statistics. SIAM Journal on Imaging Sciences 10, 416–448 (2017).
[LB19]
[LS91]
Y. Liu and C. Storey. Efficient generalized conjugate gradient algorithms, part 1: Theory. Journal of Optimization Theory and Applications 69, 129–137 (1991).
[Ngu23]
D. Nguyen. Operator-Valued Formulas for Riemannian Gradient and Hessian and Families of Tractable Metrics in Riemannian Optimization. Journal of Optimization Theory and Applications 198, 135–164 (2023), arXiv:2009.10159.
[NW06]
J. Nocedal and S. J. Wright. Numerical Optimization. Springer, New York (2006).
[Pee93]
R. Peeters. On a Riemannian version of the Levenberg-Marquardt algorithm. Technical Report 0011, VU University Amsterdam, Faculty of Economics, Business Administration and Econometrics (1993).
[PR69]
E. Polak and G. Ribière. Note sur la convergence de méthodes de directions conjuguées. Revue française d’informatique et de recherche opérationnelle 3, 35–43 (1969).
[Pol69]
B. T. Polyak. The conjugate gradient method in extremal problems. USSR Computational Mathematics and Mathematical Physics 9, 94–112 (1969).
[PN07]
T. Popiel and L. Noakes. Bézier curves and $C^2$ interpolation in Riemannian manifolds. Journal of Approximation Theory 148, 111–127 (2007).
[Pow77]
M. J. Powell. Restart procedures for the conjugate gradient method. Mathematical Programming 12, 241–254 (1977).
[SO15]
J. C. Souza and P. R. Oliveira. A proximal point algorithm for DC fuctions on Hadamard manifolds. Journal of Global Optimization 63, 797–810 (2015).
[WS22]
M. Weber and S. Sra. Riemannian Optimization via Frank-Wolfe Methods. Mathematical Programming 199, 525–556 (2022).
[ZS18]
H. Zhang and S. Sra. Towards Riemannian accelerated gradient methods, arXiv Preprint, 1806.02812 (2018).