References

[1]
S. Lipschutz. General Topology (McGraw-Hill Book Company, New York City, New York, 1965). ↩1 ↩2
[2]
S. Lang. Fundamentals of differential geometry. Vol. 191 (Springer Science & Business Media, 2012). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6
[3]
S. I. Richard L. Bishop. Tensor Analysis on Manifolds (Dover Publications, Mineola, New York, 1980). ↩1 ↩2 ↩3 ↩4
[4]
S. Lang. Real and functional analysis. Vol. 142 (Springer Science & Business Media, 2012). ↩1 ↩2
[5]
M. P. Do Carmo and J. Flaherty Francis. Riemannian geometry. Vol. 2 (Springer, 1992). ↩1 ↩2
[6]
P.-A. Absil, R. Mahony and R. Sepulchre. Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. Acta Applicandae Mathematica 80, 199–220 (2004). ↩1 ↩2 ↩3 ↩4
[7]
D. D. Holm, T. Schmah and C. Stoica. Geometric mechanics and symmetry: from finite to infinite dimensions. Vol. 12 (Oxford University Press, Oxford, UK, 2009). ↩1
[8]
P.-A. Absil, R. Mahony and R. Sepulchre. Optimization algorithms on matrix manifolds (Princeton University Press, Princeton, New Jersey, 2008). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6
[9]
T. Bendokat, R. Zimmermann and P.-A. Absil. A Grassmann manifold handbook: Basic geometry and computational aspects, arXiv preprint arXiv:2011.13699 (2020). ↩1 ↩2 ↩3 ↩4 ↩5
[10]
B. Brantner. Generalizing Adam To Manifolds For Efficiently Training Transformers, arXiv preprint arXiv:2305.16901 (2023). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6
[11]
E. Hairer, C. Lubich and G. Wanner. Geometric Numerical integration: structure-preserving algorithms for ordinary differential equations (Springer, Heidelberg, 2006). ↩1 ↩2 ↩3
[12]
T. Bendokat and R. Zimmermann. The real symplectic Stiefel and Grassmann manifolds: metrics, geodesics and applications, arXiv preprint arXiv:2108.12447 (2021). ↩1 ↩2 ↩3 ↩4
[13]
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 ↩3
[14]
B. O'neill. Semi-Riemannian geometry with applications to relativity (Academic press, New York City, New York, 1983). ↩1 ↩2
[15]
E. Celledoni and A. Iserles. Approximating the exponential from a Lie algebra to a Lie group. Mathematics of Computation 69, 1457–1480 (2000). ↩1
[16]
C. Fraikin, K. Hüper and P. V. Dooren. Optimization over the Stiefel manifold. In: PAMM: Proceedings in Applied Mathematics and Mechanics, Vol. 7 no. 1 (Wiley Online Library, 2007); pp. 1062205–1062206. ↩1
[17]
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). ↩1
[18]
N. J. Higham. The scaling and squaring method for the matrix exponential revisited. SIAM Journal on Matrix Analysis and Applications 26, 1179–1193 (2005). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7 ↩8 ↩9
[19]
A. H. Al-Mohy and N. J. Higham. A new scaling and squaring algorithm for the matrix exponential. SIAM Journal on Matrix Analysis and Applications 31, 970–989 (2010). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7
[20]
B. Skaflestad and W. M. Wright. The scaling and modified squaring method for matrix functions related to the exponential. Applied Numerical Mathematics 59, 783–799 (2009). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6
[21]
M. Hochbruck and A. Ostermann. Exponential integrators. Acta Numerica 19, 209–286 (2010). ↩1 ↩2
[22]
R. A. Horn and C. R. Johnson. Matrix analysis. 2 Edition (Cambridge University Press, Cambridge, UK, 2012). ↩1
[23]
L. N. Trefethen and M. Embree. Spectra and pseudospectra: the behavior of nonnormal matrices and operators (Princeton University Press, Princeton, NJ, 2005). ↩1
[24]
C. Moler and C. Van Loan. Nineteen dubious ways to compute the exponential of a matrix. SIAM Review 20, 801–836 (1978). ↩1
[25]
C. Moler and C. Van Loan. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Review 45, 3–49 (2003). ↩1
[26]
N. J. Higham. Functions of matrices: theory and computation (SIAM, Philadelphia, PA, 2008). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7
[27]
G. Schulz. Iterative Berechnung der reziproken Matrix. ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik 13, 57–59 (1933). ↩1 ↩2 ↩3
[28]
R. B. Sidje. Expokit: A software package for computing matrix exponentials. ACM Transactions on Mathematical Software 24, 130–156 (1998). ↩1
[29]
M. Schlarb. Covariant Derivatives on Homogeneous Spaces: Horizontal Lifts and Parallel Transport. The Journal of Geometric Analysis 34, 1–43 (2024). ↩1
[30]
I. Goodfellow, Y. Bengio and A. Courville. Deep learning (MIT press, Cambridge, MA, 2016). ↩1 ↩2
[31]
L. Kong, Y. Wang and M. Tao. Momentum stiefel optimizer, with applications to suitably-orthogonal attention, and optimal transport, arXiv preprint arXiv:2205.14173v3 (2023). ↩1
[32]
J. Li, F. Li and S. Todorovic. Efficient Riemannian Optimization on the Stiefel Manifold via the Cayley Transform. In: International Conference on Learning Representations (2020), arXiv:2002.01113. ↩1 ↩2 ↩3 ↩4
[33]
J. Nocedal and S. J. Wright. Numerical optimization (Springer, New York, NY, 2006). Second Edition. ↩1 ↩2 ↩3 ↩4 ↩5 ↩6 ↩7
[34]
I. Loshchilov and F. Hutter. Decoupled Weight Decay Regularization. In: International Conference on Learning Representations (2019). ↩1 ↩2 ↩3 ↩4 ↩5 ↩6
[35]
Y. Gu and Z. Xie. Mano: Restriking Manifold Optimization for LLM Training, arXiv preprint arXiv:2601.23000 (2026). ↩1
[36]
B. Gao, N. T. Son and T. Stykel. Optimization on the symplectic Stiefel manifold: SR decomposition-based retraction and applications. Linear Algebra and its Applications 682, 50–85 (2024). ↩1
[37]
F. Mezzadri. How to generate random matrices from the classical compact groups, arXiv preprint math-ph/0609050 (2006). ↩1
[38]
M. J. Kochenderfer and T. A. Wheeler. Algorithms for optimization (Mit Press, 2019). ↩1
[39]
T. Frankel. The geometry of physics: an introduction (Cambridge university press, Cambridge, UK, 2011).
[40]
A. Salam, E. Al-Aidarous and A. El Farouk. Optimal symplectic Householder transformations for SR decomposition. Linear Algebra and its Applications 429, 1334–1353 (2008).