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On matrix exponentials and their approximations related to optimization on the Stiefel manifold. (English) Zbl 07078371
Summary: In this paper we probe into the geometric properties of matrix exponentials and their approximations related to optimization on the Stiefel manifold. The relation between matrix exponentials and geodesics or retractions on the Stiefel manifold is discussed. Diagonal Padé approximation to matrix exponentials is used to construct new retractions. A Krylov subspace implementation of the new retractions is also considered for the orthogonal group and the Stiefel manifold with a very high rank.
MSC:
90C Mathematical programming
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[1] Absil, P.-A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton, NJ (2008) · Zbl 1147.65043
[2] Absil, P-A; Malick, J., Projection-like retractions on matrix manifolds, SIAM J. Optim., 22, 135-158, (2012) · Zbl 1248.49055
[3] Baker Jr., G.A.: Essentials of Padé Approximants. Academic Press, London (1975) · Zbl 0315.41014
[4] Barzilai, J.; Borwein, JM, Two-point step size gradient methods, IMA J. Numer. Anal., 8, 141-148, (1988) · Zbl 0638.65055
[5] do Carmo, M.P.: Riemannian Geometry. Birkhäuser Boston Inc., Boston, MA (1992). (Translated from the second Portuguese edition by Francis Flaherty. Mathematics: Theory & Applications) · Zbl 0752.53001
[6] Edelman, A.; Arias, TA; Smith, ST, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20, 303-353, (1999) · Zbl 0928.65050
[7] Gawlik, ES; Leok, M., High-order retractions on matrix manifolds using projected polynomials, SIAM J. Matrix Anal. Appl., 39, 801-828, (2018) · Zbl 1391.65105
[8] Higham, NJ, The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26, 1179-1193, (2005) · Zbl 1081.65037
[9] Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia, PA (2008) · Zbl 1167.15001
[10] Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, New York, NY (2013) · Zbl 1267.15001
[11] Huang, W.; Gallivan, KA; Absil, P-A, A Broyden class of quasi-Newton methods for Riemannian optimization, SIAM J. Optim., 25, 1660-1685, (2015) · Zbl 06479832
[12] Huang, W.; Absil, P-A; Gallivan, KA, Intrinsic representation of tangent vectors and vector transports on matrix manifolds, Numer. Math., 136, 523-543, (2017) · Zbl 1366.65062
[13] Huang, W.; Absil, P-A; Gallivan, KA, A Riemannian BFGS method without differentiated retraction for nonconvex optimization problems, SIAM J. Optim., 28, 470-495, (2018) · Zbl 1382.65177
[14] Jiang, B.; Dai, Y., A framework of constraint preserving update schemes for optimization on Stiefel manifold, Math. Program., 153, 535-575, (2015) · Zbl 1325.49037
[15] Moler, CB; Loan, CF, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45, 3-49, (2003) · Zbl 1030.65029
[16] Nishimori, Y.; Akaho, S., Learning algorithms utilizing quasi-geodesic flows on the Stiefel manifold, Neurocomputing, 67, 106-135, (2005)
[17] Ring, W.; Wirth, B., Optimization methods on Riemannian manifolds and their application to shape space, SIAM J. Optim., 22, 596-627, (2012) · Zbl 1250.90111
[18] Sato, H., A Dai-Yuan-type Riemannian conjugate gradient method with the weak Wolfe conditions, Comput. Optim. Appl., 64, 101-118, (2016) · Zbl 1338.65164
[19] Sato, H.; Iwai, T., A new, globally convergent Riemannian conjugate gradient method, Optimization, 64, 1011-1031, (2015) · Zbl 1311.65072
[20] Wen, Z.; Yin, W., A feasible method for optimization with orthogonality constraints, Math. Program., 142, 397-434, (2013) · Zbl 1281.49030
[21] Yuan, Y., Subspace methods for large scale nonlinear equations and nonlinear least squares, Optim. Eng., 10, 207-218, (2009) · Zbl 1171.65040
[22] Zhu, X., A Riemannian conjugate gradient method for optimization on the Stiefel manifold, Comput. Optim. Appl., 67, 73-110, (2017) · Zbl 1401.90230
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