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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.

[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/. ↩1 ↩2 ↩3 ↩4 ↩5
[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. ↩1 ↩2 ↩3 ↩4
[AM12]
P.-A. Absil and J. Malick. Projection-like Retractions on Matrix Manifolds. SIAM Journal on Optimization 22, 135–158 (2012). ↩1 ↩2
[AO14]
P.-A. Absil and I. V. Oseledets. Low-rank retractions: a survey and new results. Computational Optimization and Applications 62, 5–29 (2014). ↩1 ↩2
[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. ↩1
[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). ↩1
[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). ↩1
[Axl24]
S. Axler. Linear Algebra Done Right. Fourth Edition, Undergraduate Texts in Mathematics (Springer Cham, 2024). ↩1 ↩2 ↩3
[AJLS17]
N. Ay, J. Jost, H. V. Lê and L. Schwachhöfer. Information Geometry (Springer Cham, 2017). ↩1 ↩2
[Bac14]
M. Bačák. Computing medians and means in Hadamard spaces. SIAM Journal on Optimization 24, 1542–1566 (2014), arXiv:1210.2145. ↩1
[Ben21]
T. Bendokat. The Classical and Symplectic Stiefel and Grassmann Manifolds: Geometry and Applications. Ph.D. Thesis, University of Southern Denmark (2021). ↩1
[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. ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10 ↩11 ↩12 ↩13 ↩14 ↩15
[BZA20]
T. Bendokat, R. Zimmermann and P.-A. Absil. A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects, arXiv Preprint (2020), arXiv:2011.13699. ↩1 ↩2 ↩3
[BZA24]
T. Bendokat, R. Zimmermann and P.-A. Absil. A Grassmann manifold handbook: basic geometry and computational aspects. Advances in Computational Mathematics 50 (2024). ↩1 ↩2 ↩3 ↩4
[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. ↩1 ↩2
[BPS15]
[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). ↩1 ↩2 ↩3 ↩4
[BP08]
E. Binz and S. Pods. The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization (American Mathematical Society, 2008). ↩1
[BCC20]
P. Birtea, I. Caçu and D. Comănescu. Optimization on the real symplectic group. Monatshefte für Mathematik 191, 465–485 (2020). ↩1 ↩2
[BCC??]
P. Birtea, I. Caşu and D. Comănescu. First order optimality conditions and steepest descent algorithm on orthogonal Stiefel manifolds. 13, 1773–1791. ↩1
[Bou23]
[BST03]
L. J. Boya, E. Sudarshan and T. Tilma. Volumes of compact manifolds. Reports on Mathematical Physics 52, 401–422 (2003). ↩1 ↩2 ↩3
[BP19]
A. L. Brigant and S. Puechmorel. Approximation of Densities on Riemannian Manifolds. Entropy 21, 43 (2019). ↩1
[CE08]
J. Cheeger and D. G. Ebin. Comparison Theorems in Riemannian Geometry (American Mathematical Society, Providence, R.I, 2008). ↩1 ↩2
[CLLD22]
E. Chevallier, D. Li, Y. Lu and D. B. Dunson. Exponential-wrapped distributions on symmetric spaces. ArXiv Preprint (2022). ↩1 ↩2 ↩3 ↩4 ↩5
[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). ↩1
[Chi03]
Y. Chikuse. Statistics on Special Manifolds (Springer New York, 2003). ↩1 ↩2 ↩3 ↩4
[Chi12]
G. S. Chirikjian. Stochastic Models, Information Theory, and Lie Groups, Volume 2. 1 Edition, Vol. 2 of Applied and Numerical Harmonic Analysis (Birkhäuser Boston, MA, 2012). ↩1
[CB23]
C. Criscitiello and N. Boumal. An Accelerated First-Order Method for Non-convex Optimization on Manifolds. Foundations of Computational Mathematics 23, 1433–1509 (2023). ↩1
[Dav15]
R. Davis. Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices. The Electronic Journal of Combinatorics 22, P2.17 (2015). ↩1
[Dev86]
L. Devroye. Non-Uniform Random Variate Generation (Springer New York, NY, 1986). ↩1
[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. ↩1 ↩2
[Dou20]
[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. ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8
[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:physics/9806030. ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7
[FdHDF19]
L. Falorsi, P. de Haan, T. R. Davidson and P. Forré. Reparameterizing Distributions on Lie Groups. In: Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, Vol. 89 of Proceedings of Machine Learning Research (PMLR, 2019); pp. 3244–3253. ↩1 ↩2
[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). ↩1 ↩2
[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). ↩1
[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). ↩1
[GQ20]
[GSAS21a]
B. Gao, N. T. Son, P.-A. Absil and T. Stykel. Geometry of the Symplectic Stiefel Manifold Endowed with the Euclidean Metric. In: Geometric Science of Information, Vol. 12829 of Lecture Notes in Computer Science (Springer Cham, 2021); pp. 789–796. ↩1
[GSAS21b]
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). ↩1 ↩2
[Ge14]
[Gen17]
J. E. Gentle. Matrix Algebra: Theory, Computations and Applications in Statistics. 2 Edition, Springer Texts in Statistics (Springer Cham, 2017). ↩1 ↩2 ↩3
[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. ↩1 ↩2
[Hac19]
W. Hackbusch. Tensor Spaces and Numerical Tensor Calculus. 2 Edition, Vol. 56 of Springer Series in Computational Mathematics (Springer International Publishing, 2019). ↩1
[Hal15]
B. C. Hall. Lie Groups, Lie Algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics (Springer International Publishing, 2015). ↩1 ↩2 ↩3
[HMJG21]
A. Han, B. Mishra, P. Jawanpuria and J. Gao. Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry, arXiv Preprint (2021), arXiv:2110.10464. ↩1
[HMJG23]
A. Han, B. Mishra, P. Jawanpuria and J. Gao. Learning with Symmetric Positive Definite Matrices via Generalized Bures-Wasserstein Geometry. In: Geometric Science of Information, Vol. 14071 of Lecture Notes in Computer Science, edited by F. Nielsen and F. Barbaresco (Springer Nature Switzerland, 2023); pp. 405–415. ↩1 ↩2 ↩3
[HM94]
U. Helmke and J. B. Moore. Optimization and Dynamical Systems. Communications and Control Engineering (Springer London, 1994). ↩1
[HU17]
S. Hosseini and A. Uschmajew. A Riemannian Gradient Sampling Algorithm for Nonsmooth Optimization on Manifolds. SIAM J. Optim. 27, 173–189 (2017). ↩1
[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). ↩1
[HZ23]
Z. Huang and L. Zheng. Target detection based on generalized Bures–Wasserstein distance. EURASIP Journal on Advances in Signal Processing 2023 (2023). ↩1
[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). ↩1
[JSdVV26]
S. Jacobsson, L. Swijsen, J. V. der Veken and N. Vannieuwenhoven. Warped Geometries of Segre–Veronese Manifolds. SIAM Journal on Matrix Analysis and Applications 47, 1551–1577 (2026). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10
[JZ26]
R. Jensen and R. Zimmermann. A new polar factor retraction on the Stiefel manifold with closed-form inverse, arXiv Preprint, 2602.19923 (2026), arXiv:2602.19923. ↩1 ↩2 ↩3 ↩4 ↩5
[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. ↩1 ↩2 ↩3
[Joy10]
[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). ↩1
[Kar77]
H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics 30, 509–541 (1977). ↩1
[Ken84]
D. G. Kendall. Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces. Bulletin of the London Mathematical Society 16, 81–121 (1984). ↩1 ↩2 ↩3 ↩4
[Ken89]
D. G. Kendall. A Survey of the Statistical Theory of Shape. Statistical Science 4, 87–99 (1989). ↩1 ↩2 ↩3 ↩4 ↩5
[KKM22]
R. Khouja, H. Khalil and B. Mourrain. Riemannian Newton Optimization Methods for the Symmetric Tensor Approximation Problem. Linear Algebra and its Applications 637, 175–211 (2022), arXiv:2003.01469. ↩1
[KA02]
A. V. Knyazev and M. E. Argentati. Principal Angles between Subspaces in an $A$-Based Scalar Product: Algorithms and Perturbation Estimates. SIAM Journal on Scientific Computing 23, 2009–2041 (2002). ↩1 ↩2
[KL10]
O. Koch and C. Lubich. Dynamical Tensor Approximation. SIAM Journal on Matrix Analysis and Applications 31, 2360–2375 (2010). ↩1
[KSV13]
D. Kressner, M. Steinlechner and B. Vandereycken. Low-rank tensor completion by Riemannian optimization. BIT Numerical Mathematics 54, 447–468 (2013). ↩1 ↩2 ↩3
[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). ↩1
[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). ↩1
[Lee19]
J. M. Lee. Introduction to Riemannian Manifolds (Springer Cham, 2019). ↩1 ↩2 ↩3
[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. ↩1 ↩2 ↩3 ↩4
[MMP18]
L. Malagó, L. Montrucchio and G. Pistone. Wasserstein Riemannian geometry of Gaussian densities. Information Geometry 1, 137–179 (2018). ↩1
[Mar72]
G. Marsaglia. Choosing a Point from the Surface of a Sphere. Annals of Mathematical Statistics 43, 645–646 (1972). ↩1
[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. ↩1 ↩2
[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). ↩1
[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. ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10
[Pen06]
X. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision 25, 127–154 (2006). ↩1
[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. ↩1
[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. ↩1 ↩2 ↩3 ↩4 ↩5
[Sas58]
S. Sasaki. On the differential geometry of tangent bundles of Riemannian manifolds. Tohoku Math. J. 10 (1958). ↩1
[SA23]
B. Shustin and H. Avron. Riemannian optimization with a preconditioning scheme on the generalized Stiefel manifold. Journal of Computational and Applied Mathematics 423, 114953 (2023). ↩1 ↩2 ↩3 ↩4 ↩5
[SDA21]
J. Solà, J. Deray and D. Atchuthan. A micro Lie theory for state estimation in robotics (Dec 2021), arXiv:1812.01537 [cs.RO], arXiv: 1812.01537. ↩1 ↩2 ↩3 ↩4
[SK16]
A. Srivastava and E. P. Klassen. Functional and Shape Data Analysis (Springer New York, 2016). ↩1 ↩2
[dSLCY25]
T. de Surrel, F. Lotte, S. Chevallier and F. Yger. Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices. In: Proceedings of the 42nd International Conference on Machine Learning, Vol. 267 of Proceedings of Machine Learning Research (PMLR, 2025); pp. 12816–12839. ↩1
[Tal26]
S. Talaganis. Drift-Adapted Lattice Geodesics for Quantum Gate Synthesis: Exact Global Optima for Full-Isotropic SU(n) and Weighted Commuting Sectors. Universe 12, 219 (2026). ↩1
[TP23]
Y. Thanwerdas and X. Pennec. O(n)-invariant Riemannian metrics on SPD matrices. Linear Algebra and its Applications 661, 163–201 (2023). ↩1 ↩2 ↩3 ↩4 ↩5
[Tor20]
S. Tornier. Haar Measures (2020). ↩1
[TD17]
R. Tron and K. Daniilidis. The Space of Essential Matrices as a Riemannian Quotient Manifold. SIAM J. Imaging Sci. 10, 1416–1445 (2017). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9 ↩10 ↩11
[UM14]
K. Usevich and I. Markovsky. Optimization on a Grassmann manifold with application to system identification. Automatica 50, 1656–1662 (2014). ↩1
[Van13]
B. Vandereycken. Low-rank matrix completion by Riemannian optimization. SIAM Journal on Optimization 23, 1214–1236 (2013). ↩1 ↩2
[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). ↩1
[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). ↩1 ↩2
[YWL21]
K. Ye, K. S.-W. Wong and L.-H. Lim. Optimization on flag manifolds. Mathematical Programming 194, 621–660 (2021). ↩1 ↩2
[Zhu16]
X. Zhu. A Riemannian conjugate gradient method for optimization on the Stiefel manifold. Computational Optimization and Applications 67, 73–110 (2016). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6
[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). ↩1
[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. ↩1 ↩2
[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. ↩1 ↩2 ↩3 ↩4
[ZS24]
[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. ↩1 ↩2 ↩3
[ZS03]
K. Życzkowski and H.-J. Sommers. Hilbert–Schmidt volume of the set of mixed quantum states. Journal of Physics A: Mathematical and General 36, 10115–10130 (2003). ↩1 ↩2 ↩3 ↩4