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.

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.
P.-A. Absil, R. Mahony and R. Sepulchre. Optimization Algorithms on Matrix Manifolds (Princeton University Press, 2008), available online at
P.-A. Absil and J. Malick. Projection-like Retractions on Matrix Manifolds. SIAM Journal on Optimization 22, 135–158 (2012).
P.-A. Absil and I. V. Oseledets. Low-rank retractions: a survey and new results. Computational Optimization and Applications 62, 5–29 (2014).
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.
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).
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.
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).
N. Ay, J. Jost, H. V. Lê and L. Schwachhöfer. Information Geometry (Springer Cham, 2017).
M. Bačák. Computing medians and means in Hadamard spaces. SIAM Journal on Optimization 24, 1542–1566 (2014), arXiv:1210.2145.
T. Bendokat and R. Zimmermann. The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications, arXiv Preprint, 2108.12447 (2021), arXiv:2108.12447.
T. Bendokat, R. Zimmermann and P.-A. Absil. A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects, arXiv Preprint (2020), arXiv:2011.13699.
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.
E. Biny and S. Pods. The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization (American Mathematical Socienty, 2008).
P. Birtea, I. Caçu and D. Comănescu. Optimization on the real symplectic group. Monatshefte für Mathematik 191, 465–485 (2020).
L. J. Boya, E. Sudarshan and T. Tilma. Volumes of compact manifolds. Reports on Mathematical Physics 52, 401–422 (2003).
A. L. Brigant and S. Puechmorel. Approximation of Densities on Riemannian Manifolds. Entropy 21, 43 (2019).
J. Cheeger and D. G. Ebin. Comparison Theorems in Riemannian Geometry (American Mathematical Society, Providence, R.I, 2008).
E. Chevallier, D. Li, Y. Lu and D. B. Dunson. Exponential-wrapped distributions on symmetric spaces. ArXiv Preprint (2022).
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).
Y. Chikuse. Statistics on Special Manifolds (Springer New York, 2003).
L. Devroye. Non-Uniform Random Variate Generation (Springer New York, NY, 1986).
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.
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.
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.
L. Falorsi, P. de Haan, T. R. Davidson and P. Forré. Reparameterizing Distributions on Lie Groups, arXiv Preprint (2019).
S. Fiori. Solving Minimal-Distance Problems over the Manifold of Real-Symplectic Matrices. SIAM Journal on Matrix Analysis and Applications 32, 938–968 (2011).
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).
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).
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).
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.
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.
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.
S. Hosseini and A. Uschmajew. A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds. SIAM J. Optim. 27, 173–189 (2017).
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).
K. Hüper, I. Markina and F. S. Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics 13, 55 (2021).
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.
T. Kaneko, S. Fiori and T. Tanaka. Empirical Arithmetic Averaging Over the Compact Stiefel Manifold. IEEE Transactions on Signal Processing 61, 883–894 (2013).
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30, 509–541 (1977).
D. G. Kendall. Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces. Bulletin of the London Mathematical Society 16, 81–121 (1984).
D. G. Kendall. A Survey of the Statistical Theory of Shape. Statistical Sciences 4, 87–99 (1989).
O. Koch and C. Lubich. Dynamical Tensor Approximation. SIAM Journal on Matrix Analysis and Applications 31, 2360–2375 (2010).
D. Kressner, M. Steinlechner and B. Vandereycken. Low-rank tensor completion by Riemannian optimization. BIT Numerical Mathematics 54, 447–468 (2013).
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).
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).
J. M. Lee. Introduction to Riemannian Manifolds (Springer Cham, 2019).
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.
L. Malagó, L. Montrucchio and G. Pistone. Wasserstein Riemannian geometry of Gaussian densities. Information Geometry 1, 137–179 (2018).
G. Marsaglia. Choosing a Point from the Surface of a Sphere. Annals of Mathematical Statistics 43, 645–646 (1972).
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:
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).
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.
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.
X. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision 25, 127–154 (2006).
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.
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.
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.
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).
S. Sasaki. On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10 (1958).
A. Srivastava and E. P. Klassen. Functional and Shape Data Analysis (Springer New York, 2016).
E. Suhubi. Exterior Analysis: Using Applications of Differential Forms (Academic Press, 2013).
S. Tornier. Haar Measures (2020).
R. Tron and K. Daniilidis. The Space of Essential Matrices as a Riemannian Quotient Manifold. SIAM J. Imaging Sci. 10, 1416–1445 (2017).
B. Vandereycken. Low-rank matrix completion by Riemannian optimization. SIAM Journal on Optimization 23, 1214–1236 (2013).
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).
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).
K. Ye, K. S.-W. Wong and L.-H. Lim. Optimization on flag manifolds. Mathematical Programming 194, 621–660 (2021).
X. Zhu. A Riemannian conjugate gradient method for optimization on the Stiefel manifold. Computational Optimization and Applications 67, 73–110 (2016).
X. Zhu and C. Duan. On matrix exponentials and their approximations related to optimization on the Stiefel manifold. Optimization Letters 13, 1069–1083 (2018).
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.
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.
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.