Random orthogonal matrices and the Cayley transform. (English) Zbl 1466.60010

Summary: Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices are complicated by their constrained support. Accordingly, we parametrize the Stiefel and Grassmann manifolds, represented as subsets of orthogonal matrices, in terms of Euclidean parameters using the Cayley transform. We derive the necessary Jacobian terms for change of variables formulas. Given a density defined on the Stiefel or Grassmann manifold, these allow us to specify the corresponding density for the Euclidean parameters, and vice versa. As an application, we present a Markov chain Monte Carlo approach to simulating from distributions on the Stiefel and Grassmann manifolds. Finally, we establish that the Euclidean parameters corresponding to a uniform orthogonal matrix can be approximated asymptotically by independent normals. This result contributes to the growing literature on normal approximations to the entries of random orthogonal matrices or transformations thereof.


60B20 Random matrices (probabilistic aspects)
60F05 Central limit and other weak theorems
62R30 Statistics on manifolds


Stan; PyMC; rstiefel
Full Text: DOI arXiv Euclid


[1] Anderson, T.W., Olkin, I. and Underhill, L.G. (1987). Generation of random orthogonal matrices. SIAM J. Sci. Statist. Comput. 8 625-629. · Zbl 0637.65004
[2] Borel, É. (1906). Sur les principes de la théorie cinétique des gaz. Ann. Sci. Éc. Norm. Supér. (3) 23 9-32. · JFM 37.0944.01
[3] Bourgade, P., Nikeghbali, A. and Rouault, A. (2007). Hua-Pickrell measures on general compact groups. Available at arXiv:0712.0848v1. · Zbl 1123.60004
[4] Bourgade, P., Nikeghbali, A. and Rouault, A. (2011). Ewens measures on compact groups and hypergeometric kernels. In Séminaire de Probabilités XLIII. Lecture Notes in Math. 2006 351-377. Berlin: Springer. · Zbl 1228.60011
[5] Byrne, S. and Girolami, M. (2013). Geodesic Monte Carlo on embedded manifolds. Scand. J. Stat. 40 825-845. · Zbl 1349.62186
[6] Carpenter, B., Gelman, A., Hoffman, M.D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P. and Riddell, A. (2017). Stan: A probabilistic programming language. J. Stat. Softw. 76 1-32.
[7] Cayley, A. (1846). Sur quelques propriétés des déterminants gauches. J. Reine Angew. Math. 32 119-123. · ERAM 032.0912cj
[8] Chatterjee, S. and Meckes, E. (2008). Multivariate normal approximation using exchangeable pairs. ALEA Lat. Am. J. Probab. Math. Stat. 4 257-283. · Zbl 1162.60310
[9] Chikuse, Y. (2003). Statistics on Special Manifolds. Lecture Notes in Statist. 174. New York: Springer. · Zbl 1026.62051
[10] Cook, R.D., Li, B. and Chiaromonte, F. (2010). Envelope models for parsimonious and efficient multivariate linear regression. Statist. Sinica 20 927-960. · Zbl 1259.62059
[11] Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J. and Knuth, D.E. (1996). On the Lambert \(W\) function. Adv. Comput. Math. 5 329-359. · Zbl 0863.65008
[12] Cron, A. and West, M. (2016). Models of random sparse eigenmatrices and Bayesian analysis of multivariate structure. In Statistical Analysis for High-Dimensional Data. Abel Symp. 11 125-153. Cham: Springer. · Zbl 1381.62127
[13] D’Aristotile, A., Diaconis, P. and Newman, C.M. (2003). Brownian motion and the classical groups. In Probability, Statistics and Their Applications: Papers in Honor of Rabi Bhattacharya. Institute of Mathematical Statistics Lecture Notes—Monograph Series 41 97-116. Beachwood, OH: IMS. · Zbl 1056.60081
[14] Diaconis, P. and Forrester, P.J. (2017). Hurwitz and the origins of random matrix theory in mathematics. Random Matrices Theory Appl. 6 Art. ID 1730001. · Zbl 1398.11119
[15] Diaconis, P. and Freedman, D. (1987). A dozen de Finetti-style results in search of a theory. Ann. Inst. Henri Poincaré Probab. Stat. 23 397-423. · Zbl 0619.60039
[16] Diaconis, P., Holmes, S. and Shahshahani, M. (2013). Sampling from a manifold. In Advances in Modern Statistical Theory and Applications: A Festschrift in Honor of Morris L. Eaton. Inst. Math. Stat. (IMS) Collect. 10 102-125. Beachwood, OH: IMS. · Zbl 1356.62015
[17] Diaconis, P., Seiler, C. and Holmes, S. (2014). Connections and extensions: A discussion of the paper by Girolami and Byrne. Scand. J. Stat. 41 3-7. · Zbl 1349.62192
[18] Diaconis, P. and Shahshahani, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31 49-62. · Zbl 0807.15015
[19] Diaconis, P.W., Eaton, M.L. and Lauritzen, S.L. (1992). Finite de Finetti theorems in linear models and multivariate analysis. Scand. J. Stat. 19 289-315. · Zbl 0795.62049
[20] Eaton, M.L. (1983). Multivariate Statistics: A Vector Space Approach. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
[21] Eaton, M.L. (1989). Group Invariance Applications in Statistics. NSF-CBMS Regional Conference Series in Probability and Statistics 1. Hayward, CA: IMS; Alexandria, VA: Amer. Statist. Assoc.
[22] Edelman, A., Arias, T.A. and Smith, S.T. (1999). The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20 303-353. · Zbl 0928.65050
[23] Grayson, M.A. (1989). A short note on the evolution of a surface by its mean curvature. Duke Math. J. 58 555-558. · Zbl 0677.53059
[24] Hoff, P.D. (2007). Model averaging and dimension selection for the singular value decomposition. J. Amer. Statist. Assoc. 102 674-685. · Zbl 1172.62318
[25] Hoff, P.D. (2009). Simulation of the matrix Bingham-von Mises-Fisher distribution, with applications to multivariate and relational data. J. Comput. Graph. Statist. 18 438-456.
[26] Hubbard, J.H. and Hubbard, B.B. (2009). Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, 5th ed. Ithaca, NY: Matrix Editions. · Zbl 0918.00001
[27] Hurwitz, A. (1897). Über die Erzeugung der invarianten durch integration. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1897 71-90. · JFM 28.0103.03
[28] James, A.T. (1954). Normal multivariate analysis and the orthogonal group. Ann. Math. Stat. 25 40-75. · Zbl 0055.13203
[29] Jauch, M., Hoff, P.D. and Dunson, D.B. (2019). Monte Carlo simulation on the Stiefel manifold via polar expansion. Available at arXiv:1906.07684. · Zbl 07499906
[30] Jiang, T. (2006). How many entries of a typical orthogonal matrix can be approximated by independent normals? Ann. Probab. 34 1497-1529. · Zbl 1107.15018
[31] Jiang, T. and Ma, Y. (2019). Distances between random orthogonal matrices and independent normals. Trans. Amer. Math. Soc. 372 1509-1553. · Zbl 1417.15051
[32] Johansson, K. (1997). On random matrices from the compact classical groups. Ann. of Math. (2) 145 519-545. · Zbl 0883.60010
[33] Johnstone, I.M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. · Zbl 1016.62078
[34] Keating, J.P. and Snaith, N.C. (2000). Random matrix theory and \(\zeta(1/2+it)\). Comm. Math. Phys. 214 57-89. · Zbl 1051.11048
[35] León, C.A., Massé, J.-C. and Rivest, L.-P. (2006). A statistical model for random rotations. J. Multivariate Anal. 97 412-430. · Zbl 1085.62066
[36] Magnus, J.R. (1988). Linear Structures. Griffin’s Statistical Monographs & Courses 42. London: Charles Griffin & Co., Ltd.; New York: The Clarendon Press.
[37] Magnus, J.R. and Neudecker, H. (1979). The commutation matrix: Some properties and applications. Ann. Statist. 7 381-394. · Zbl 0414.62040
[38] Magnus, J.R. and Neudecker, H. (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. Wiley Series in Probability and Statistics. Chichester: Wiley. · Zbl 0651.15001
[39] Mardia, K.V. and Jupp, P.E. (2009). Directional Statistics. Wiley Series in Probability and Statistics. Chichester: Wiley.
[40] Maxwell, J.C. (1875). Theory of Heat, 4th ed. London: Longmans.
[41] Maxwell, J.C. (1878). On Boltzmann’s theorem on the average distribution of energy in a system of material points. Trans. Camb. Philos. Soc. 12 547-575.
[42] Meckes, E. (2008). Linear functions on the classical matrix groups. Trans. Amer. Math. Soc. 360 5355-5366. · Zbl 1149.60017
[43] Mehler, F.G. (1866). Ueber die Entwicklung einer Function von beliebig vielen Variablen nach Laplaceschen Functionen höherer Ordnung. J. Reine Angew. Math. 66 161-176. · ERAM 066.1720cj
[44] Neal, R.M. (2011). MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo. Chapman & Hall/CRC Handb. Mod. Stat. Methods 113-162. Boca Raton, FL: CRC Press. · Zbl 1229.65018
[45] Neudecker, H. (1983). On Jacobians of transformations with skew-symmetric, strictly (lower) triangular or diagonal matrix arguments. Linear Multilinear Algebra 14 271-295. · Zbl 0528.15005
[46] Pourzanjani, A.A., Jiang, R.M., Mitchell, B., Atzberger, P.J. and Petzold, L.R. (2017). General Bayesian inference over the Stiefel manifold via the givens transform. Available at arXiv:1710.09443v2. · Zbl 1493.62133
[47] Rains, E.M. (1997). High powers of random elements of compact Lie groups. Probab. Theory Related Fields 107 219-241. · Zbl 0868.60012
[48] Rao, V., Lin, L. and Dunson, D.B. (2016). Data augmentation for models based on rejection sampling. Biometrika 103 319-335. · Zbl 1499.62411
[49] Salvatier, J., Wiecki, T.V. and Fonnesbeck, C. (2016). Probabilistic programming in Python using PyMC3. PeerJ Comput. Sci. 2 Art. ID e55.
[50] Shepard, R., Brozell, S.R. and Gidofalvi, G. (2015). The representation and parametrization of orthogonal matrices. J. Phys. Chem. A 119 7924-7939.
[51] Stam, A.J. (1982). Limit theorems for uniform distributions on spheres in high-dimensional Euclidean spaces. J. Appl. Probab. 19 221-228. · Zbl 0485.60024
[52] Stein, C. (1995). The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices. Technical Report 470 Dept. Statistics, Stanford Univ.
[53] Stewart, K. (2019). Total variation approximation of random orthogonal matrices by Gaussian matrices. J. Theoret. Probab. · Zbl 1445.60013
[54] Traynor, T. (1994). Change of variable for Hausdorff measure (from the beginning). Rend. Istit. Mat. Univ. Trieste 26 327-347. · Zbl 0876.28010
[55] van Handel, R. (2016). Probability in high dimension. Technical report, Princeton Univ.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.