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Optimization on flag manifolds

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Abstract

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical pde; they arise in the form of Krylov subspaces in matrix computations, and as multiresolution analysis in wavelets constructions. They are common in statistics too—principal component, canonical correlation, and correspondence analyses may all be viewed as methods for extracting flags from a data set. The main goal of this article is to develop the tools needed for optimizing over a set of flags, which is a smooth manifold called the flag manifold, and it contains the Grassmannian as the simplest special case. We will derive closed-form analytic expressions for various differential geometric objects required for Riemannian optimization algorithms on the flag manifold; introducing various systems of extrinsic coordinates that allow us to parameterize points, metrics, tangent spaces, geodesics, distances, parallel transports, gradients, Hessians in terms of matrices and matrix operations; and thereby permitting us to formulate steepest descent, conjugate gradient, and Newton algorithms on the flag manifold using only standard numerical linear algebra.

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Notes

  1. Discounting manifolds that can be realized as products or open subsets of these manifolds, e.g., those considered in [1, 31, 35, 44].

  2. More precisely, the covariant derivative associated with the Levi-Civita connection on M.

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Acknowledgements

We thank the two anonymous referees for their exceptionally helpful suggestions and comments. KY is partially supported by National Key Research and Development Program of China No. 2018YFA0306702 and National Key Research and Development Program of China No. 2020YFA0712300, NSFC Grant No. 11801548 and NSFC Grant No. 11688101. LHL is supported by DARPA D15AP00109, HR00112190040, NSF IIS 1546413, DMS 1854831.

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ke Ye

  2. Department of Statistics, University of Chicago, Chicago, IL, 60637-1514, USA

    Ken Sze-Wai Wong

  3. Computational and Applied Mathematics Initiative, Department of Statistics, University of Chicago, Chicago, IL, 60637-1514, USA

    Lek-Heng Lim

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  1. Ke Ye
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KY is supported by NSFC Grant No. 11801548, NSFC Grant No. 11688101, and National Key R&D Program of China Grant No. 2018YFA0306702 and 2020YFA0712300. LHL is supported by DARPA D15AP00109 and HR00112190040, NSF IIS 1546413, DMS 1854831.

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Ye, K., Wong, K.SW. & Lim, LH. Optimization on flag manifolds. Math. Program. 194, 621–660 (2022). https://doi.org/10.1007/s10107-021-01640-3

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