Expected distances on manifolds of partially oriented flags
HTML articles powered by AMS MathViewer
- by Brenden Balch, Chris Peterson and Clayton Shonkwiler PDF
- Proc. Amer. Math. Soc. 149 (2021), 3553-3567 Request permission
Abstract:
Flag manifolds are generalizations of projective spaces and other Grassmannians: they parametrize flags, which are nested sequences of subspaces in a given vector space. These are important objects in algebraic and differential geometry, but are also increasingly being used in data science, where many types of data are properly understood as subspaces rather than vectors. In this paper we discuss partially oriented flag manifolds, which parametrize flags in which some of the subspaces may be endowed with an orientation. We compute the expected distance between random points on some low-dimensional examples, which we view as a statistical baseline against which to compare the distances between particular partially oriented flags coming from geometry or data.References
- P.-A. Absil, A. Edelman, and P. Koev, On the largest principal angle between random subspaces, Linear Algebra Appl. 414 (2006), no. 1, 288–294. MR 2209246, DOI 10.1016/j.laa.2005.10.004
- Yasuko Chikuse, Statistics on special manifolds, Lecture Notes in Statistics, vol. 174, Springer-Verlag, New York, 2003. MR 1960435, DOI 10.1007/978-0-387-21540-2
- Bruce Draper, Michael Kirby, Justin Marks, Tim Marrinan, and Chris Peterson, A flag representation for finite collections of subspaces of mixed dimensions, Linear Algebra Appl. 451 (2014), 15–32. MR 3198905, DOI 10.1016/j.laa.2014.03.022
- Alan Edelman, Tomás A. Arias, and Steven T. Smith, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl. 20 (1999), no. 2, 303–353. MR 1646856, DOI 10.1137/S0895479895290954
- Helaman R. P. Ferguson, David H. Bailey, and Steve Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comp. 68 (1999), no. 225, 351–369. MR 1489971, DOI 10.1090/S0025-5718-99-00995-3
- V. L. Girko, Distribution of eigenvalues and eigenvectors of orthogonal random matrices, Ukrain. Mat. Zh. 37 (1985), no. 5, 568–575, 676 (Russian). MR 815299
- Kee Yuen Lam, A formula for the tangent bundle of flag manifolds and related manifolds, Trans. Amer. Math. Soc. 213 (1975), 305–314. MR 431194, DOI 10.1090/S0002-9947-1975-0431194-X
- Assyr Abdulle, Simone Deparis, Daniel Kressner, Fabio Nobile, and Marco Picasso (eds.), Numerical mathematics and advanced applications—ENUMATH 2013, Lecture Notes in Computational Science and Engineering, vol. 103, Springer, Cham, 2015. MR 3523854
- Yasunori Nishimori, Shotaro Akaho, and Mark D. Plumbley, Riemannian optimization method on generalized flag manifolds for complex and subspace ICA, AIP Conf. Proc. 872 (2006), no. 1, 89–96.
- Eugene Salamin, Application of quaternions to computation with rotations, Working paper, Stanford AI Lab, 1979.
- P. Sankaran and P. Zvengrowski, Stable parallelizability of partially oriented flag manifolds, Pacific J. Math. 128 (1987), no. 2, 349–359. MR 888523, DOI 10.2140/pjm.1987.128.349
- Parameswaran Sankaran and Peter Zvengrowski, Stable parallelizability of partially oriented flag manifolds. II, Canad. J. Math. 49 (1997), no. 6, 1323–1339. MR 1611676, DOI 10.4153/CJM-1997-065-1
- Ke Ye, Ken Sze-Wai Wong, and Lek-Heng Lim, Optimization on flag manifolds, Preprint, arXiv:1907.00949 [math.OC], 2019.
Additional Information
- Brenden Balch
- Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
- MR Author ID: 1414726
- ORCID: 0000-0002-9484-8461
- Email: balch@math.colostate.edu
- Chris Peterson
- Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
- MR Author ID: 359254
- Email: peterson@math.colostate.edu
- Clayton Shonkwiler
- Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
- MR Author ID: 887567
- ORCID: 0000-0002-4811-8409
- Email: clay@shonkwiler.org
- Received by editor(s): February 3, 2020
- Received by editor(s) in revised form: October 19, 2020
- Published electronically: May 12, 2021
- Additional Notes: The second author was supported in part by NSF grants ATD #1712788 and CCF–BSF:CIF #1830676
The third author was supported in part by Simons Foundation grant #354225 - Communicated by: Dean Yang
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 149 (2021), 3553-3567
- MSC (2020): Primary 53C30; Secondary 60D05, 51N25
- DOI: https://doi.org/10.1090/proc/15521
- MathSciNet review: 4273156