GitHub

Literature

We are slowly moving to using DocumenterCitations.jl. The goal is to have all references used / mentioned in the documentation of Manifolds.jl also listed here. If you notice a reference still defined in a footnote, please change it into a BibTeX reference and open a PR

Usually you will find a small reference section at the end of every documentation page that contains references for just that page.

[AMT13]
P. -.-A. Absil, R. Mahony and J. Trumpf. An Extrinsic Look at the Riemannian Hessian. In: Geometric Science of Information, edited by F. Nielsen and F. Barbaresco (Springer Berlin Heidelberg, 2013); pp. 361–368.
[AMS08]
P.-A. Absil, R. Mahony and R. Sepulchre. Optimization Algorithms on Matrix Manifolds (Princeton University Press, 2008), available online at press.princeton.edu/chapters/absil/.
[AM12]
P.-A. Absil and J. Malick. Projection-like Retractions on Matrix Manifolds. SIAM Journal on Optimization 22, 135–158 (2012).
[AO14]
P.-A. Absil and I. V. Oseledets. Low-rank retractions: a survey and new results. Computational Optimization and Applications 62, 5–29 (2014).
[ATV13]
B. Afsari, R. Tron and R. Vidal. On the Convergence of Gradient Descent for Finding the Riemannian Center of Mass. SIAM Journal on Control and Optimization 51, 2230–2260 (2013), arXiv:1201.0925.
[AR13]
D. Andrica and R.-A. Rohan. Computing the Rodrigues coefficients of the exponential map of the Lie groups of matrices. Balkan Journal of Geometry and Its Applications 18, 1–10 (2013).
[ALRV14]
E. Andruchow, G. Larotonda, L. Recht and A. Varela. The left invariant metric in the general linear group. Journal of Geometry and Physics 86, 241–257 (2014), arXiv:1109.0520.
[ABBR23]
S. D. Axen, M. Baran, R. Bergmann and K. Rzecki. Manifolds.Jl: An Extensible Julia Framework for Data Analysis on Manifolds. ACM Transactions on Mathematical Software 49 (2023).
[AJLS17]
N. Ay, J. Jost, H. V. Lê and L. Schwachhöfer. Information Geometry (Springer Cham, 2017).
[Bac14]
M. Bačák. Computing medians and means in Hadamard spaces. SIAM Journal on Optimization 24, 1542–1566 (2014), arXiv:1210.2145.
[BZ21]
T. Bendokat and R. Zimmermann. The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications, arXiv Preprint, 2108.12447 (2021), arXiv:2108.12447.
[BZA20]
T. Bendokat, R. Zimmermann and P.-A. Absil. A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects, arXiv Preprint (2020), arXiv:2011.13699.
[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.
[BP08]
E. Biny and S. Pods. The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization (American Mathematical Socienty, 2008).
[BCC20]
P. Birtea, I. Caçu and D. Comănescu. Optimization on the real symplectic group. Monatshefte für Mathematik 191, 465–485 (2020).
[BST03]
L. J. Boya, E. Sudarshan and T. Tilma. Volumes of compact manifolds. Reports on Mathematical Physics 52, 401–422 (2003).
[BP19]
A. L. Brigant and S. Puechmorel. Approximation of Densities on Riemannian Manifolds. Entropy 21, 43 (2019).
[CE08]
J. Cheeger and D. G. Ebin. Comparison Theorems in Riemannian Geometry (American Mathematical Society, Providence, R.I, 2008).
[CLLD22]
E. Chevallier, D. Li, Y. Lu and D. B. Dunson. Exponential-wrapped distributions on symmetric spaces. ArXiv Preprint (2022).
[CKA17]
E. Chevallier, E. Kalunga and J. Angulo. Kernel Density Estimation on Spaces of Gaussian Distributions and Symmetric Positive Definite Matrices. SIAM Journal on Imaging Sciences 10, 191–215 (2017).
[Chi03]
Y. Chikuse. Statistics on Special Manifolds (Springer New York, 2003).
[Dev86]
L. Devroye. Non-Uniform Random Variate Generation (Springer New York, NY, 1986).
[DBV21]
N. Dewaele, P. Breiding and N. Vannieuwenhoven. The condition number of many tensor decompositions is invariant under Tucker compression, arXiv Preprint (2021), arXiv:2106.13034.
[DH19]
A. Douik and B. Hassibi. Manifold Optimization Over the Set of Doubly Stochastic Matrices: A Second-Order Geometry. IEEE Transactions on Signal Processing 67, 5761–5774 (2019), arXiv:1802.02628.
[EAS98]
A. Edelman, T. A. Arias and S. T. Smith. The Geometry of Algorithms with Orthogonality Constraints. SIAM Journal on Matrix Analysis and Applications 20, 303–353 (1998), arXiv:806030.
[FdHDF19]
L. Falorsi, P. de Haan, T. R. Davidson and P. Forré. Reparameterizing Distributions on Lie Groups, arXiv Preprint (2019).
[Fio11]
S. Fiori. Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices. SIAM Journal on Matrix Analysis and Applications 32, 938–968 (2011).
[FVJ08]
P. T. Fletcher, S. Venkatasubramanian and S. Joshi. Robust statistics on Riemannian manifolds via the geometric median. In: 2008 IEEE Conference on Computer Vision and Pattern Recognition (2008).
[GX02]
J. Gallier and D. Xu. Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. International Journal of Robotics and Automation 17, 1–11 (2002).
[GSAS21]
B. Gao, N. T. Son, P.-A. Absil and T. Stykel. Riemannian Optimization on the Symplectic Stiefel Manifold. SIAM Journal on Optimization 31, 1546–1575 (2021).
[Ge14]
[Gil08]
M. B. Giles. Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation. In: Advances in Automatic Differentiation, Lecture Notes in Computational Science and Engineering, edited by C. H. Bischof, H. M. Bücker, P. Hovland, U. Naumann and J. Utke (Springer, Berlin, Heidelberg, 2008); pp. 35–44.
[GMTP21]
N. Guigui, E. Maignant, A. Trouvé and X. Pennec. Parallel Transport on Kendall Shape Spaces. In: Geometric Science of Information (SPringer Cham, 2021); pp. 103–110.
[HMJG21]
A. Han, B. Mushra, P. Jawapanpuria and J. Gao. Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry, arXive preprint (2021), arXiv:2110.10464.
[HU17]
S. Hosseini and A. Uschmajew. A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds. SIAM J. Optim. 27, 173–189 (2017).
[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).
[HML21]
K. Hüper, I. Markina and F. S. Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics 13, 55 (2021).
[JBAS10]
M. Journée, F. Bach, P.-A. Absil and R. Sepulchre. Low-Rank Optimization on the Cone of Positive Semidefinite Matrices. SIAM Journal on Optimization 20, 2327–2351 (2010), arXiv:0807.4423.
[KFT13]
T. Kaneko, S. Fiori and T. Tanaka. Empirical Arithmetic Averaging Over the Compact Stiefel Manifold. IEEE Transactions on Signal Processing 61, 883–894 (2013).
[Kar77]
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30, 509–541 (1977).
[Ken84]
D. G. Kendall. Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces. Bulletin of the London Mathematical Society 16, 81–121 (1984).
[Ken89]
D. G. Kendall. A Survey of the Statistical Theory of Shape. Statistical Sciences 4, 87–99 (1989).
[KL10]
O. Koch and C. Lubich. Dynamical Tensor Approximation. SIAM Journal on Matrix Analysis and Applications 31, 2360–2375 (2010).
[KSV13]
D. Kressner, M. Steinlechner and B. Vandereycken. Low-rank tensor completion by Riemannian optimization. BIT Numerical Mathematics 54, 447–468 (2013).
[LW19]
N. Langrené and X. Warin. Fast and Stable Multivariate Kernel Density Estimation by Fast Sum Updating. Journal of Computational and Graphical Statistics 28, 596–608 (2019).
[LMV00]
L. D. Lathauwer, B. D. Moor and J. Vandewalle. A Multilinear Singular Value Decomposition. SIAM Journal on Matrix Analysis and Applications 21, 1253–1278 (2000).
[Lee19]
J. M. Lee. Introduction to Riemannian Manifolds (Springer Cham, 2019).
[Lin19]
Z. Lin. Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky Decomposition. SIAM Journal on Matrix Analysis and Applications 40, 1353–1370 (2019), arXiv:1908.09326.
[MMP18]
L. Malagó, L. Montrucchio and G. Pistone. Wasserstein Riemannian geometry of Gaussian densities. Information Geometry 1, 137–179 (2018).
[Mar72]
G. Marsaglia. Choosing a Point from the Surface of a Sphere. Annals of Mathematical Statistics 43, 645–646 (1972).
[MA20]
E. Massart and P.-A. Absil. Quotient Geometry with Simple Geodesics for the Manifold of Fixed-Rank Positive-Semidefinite Matrices. SIAM Journal on Matrix Analysis and Applications 41, 171–198 (2020). Preprint: sites.uclouvain.be/absil/2018.06.
[MF12]
P. Muralidharan and P. T. Fletcher. Sasaki metrics for analysis of longitudinal data on manifolds. In: 2012 IEEE Conference on Computer Vision and Pattern Recognition (2012).
[NM16]
P. Neff and R. J. Martin. Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics. J. Geom. Mech. 8, 323–357 (2016), arXiv:1409.7849.
[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.
[Pen06]
X. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision 25, 127–154 (2006).
[PA12]
X. Pennec and V. Arsigny. Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups. In: Matrix Information Geometry (Springer, Berlin, Heidelberg, 2012); pp. 123–166, arXiv:00699361.
[PL20]
X. Pennec and M. Lorenzi. Beyond Riemannian geometry: The affine connection setting for transformation groups. In: Riemannian Geometric Statistics in Medical Image Analysis (Elsevier, 2020); pp. 169–229.
[Ren11]
Q. Rentmeesters. A gradient method for geodesic data fitting on some symmetric Riemannian manifolds. In: IEEE Conference on Decision and Control and European Control Conference (2011); pp. 7141–7146.
[Ric88]
J. M. Rico Martinez. Representations of the Euclidean group and its applications to the kinematics of spatial chains. Ph.D. Thesis, University of FLorida (1988).
[Sas58]
S. Sasaki. On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10 (1958).
[SK16]
A. Srivastava and E. P. Klassen. Functional and Shape Data Analysis (Springer New York, 2016).
[Suh13]
E. Suhubi. Exterior Analysis: Using Applications of Differential Forms (Academic Press, 2013).
[Tor20]
S. Tornier. Haar Measures (2020).
[TD17]
R. Tron and K. Daniilidis. The Space of Essential Matrices as a Riemannian Quotient Manifold. SIAM J. Imaging Sci. 10, 1416–1445 (2017).
[Van13]
B. Vandereycken. Low-rank matrix completion by Riemannian optimization. SIAM Journal on Optimization 23, 1214–1236 (2013).
[VVM12]
N. Vannieuwenhoven, R. Vandebril and K. Meerbergen. A New Truncation Strategy for the Higher-Order Singular Value Decomposition. SIAM Journal on Scientific Computing 34, A1027–A1052 (2012).
[WSF18]
J. Wang, H. Sun and S. Fiori. A Riemannian-steepest-descent approach for optimization on the real symplectic group. Mathematical Methods in the Applied Science 41, 4273–4286 (2018).
[YWL21]
K. Ye, K. S.-W. Wong and L.-H. Lim. Optimization on flag manifolds. Mathematical Programming 194, 621–660 (2021).
[Zhu16]
X. Zhu. A Riemannian conjugate gradient method for optimization on the Stiefel manifold. Computational Optimization and Applications 67, 73–110 (2016).
[ZD18]
X. Zhu and C. Duan. On matrix exponentials and their approximations related to optimization on the Stiefel manifold. Optimization Letters 13, 1069–1083 (2018).
[Zim17]
R. Zimmermann. A Matrix-Algebraic Algorithm for the Riemannian Logarithm on the Stiefel Manifold under the Canonical Metric. SIAM J. Matrix Anal. Appl. 38, 322–342 (2017), arXiv:1604.05054.
[ZH22]
R. Zimmermann and K. Hüper. Computing the Riemannian Logarithm on the Stiefel Manifold: Metrics, Methods, and Performance. SIAM Journal on Matrix Analysis and Applications 43, 953–980 (2022), arXiv:2103.12046.
[APSS17]
F. Åström, S. Petra, B. Schmitzer and C. Schnörr. Image Labeling by Assignment. Journal of Mathematical Imaging and Vision 58, 211–238 (2017), arXiv:1603.05285.