Practical gradient and conjugate gradient methods on flag manifolds

Xiaojing Zhu () and Chungen Shen
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Xiaojing Zhu: Shanghai University of Electric Power
Chungen Shen: University of Shanghai for Science and Technology

Computational Optimization and Applications, 2024, vol. 88, issue 2, No 4, 524 pages

Abstract: Abstract Flag manifolds, sets of nested sequences of linear subspaces with fixed dimensions, are rising in numerical analysis and statistics. The current optimization algorithms on flag manifolds are based on the exponential map and parallel transport, which are expensive to compute. In this paper we propose practical optimization methods on flag manifolds without the exponential map and parallel transport. Observing that flag manifolds have a similar homogeneous structure with Grassmann and Stiefel manifolds, we generalize some typical retractions and vector transports to flag manifolds, including the Cayley-type retraction and vector transport, the QR-based and polar-based retractions, the projection-type vector transport and the projection of the differentiated polar-based retraction as a vector transport. Theoretical properties and efficient implementations of the proposed retractions and vector transports are discussed. Then we establish Riemannian gradient and Riemannian conjugate gradient algorithms based on these retractions and vector transports. Numerical results on the problem of nonlinear eigenflags demonstrate that our algorithms have a great advantage in efficiency over the existing ones.

Keywords: Manifold optimization; Flag manifold; Riemannian gradient method; Riemannian conjugate gradient method; 65K05; 90C06; 90C30; 90C48 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10589-024-00568-6

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