Transportless conjugate gradient for optimization on Stiefel manifold
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Transportless conjugate gradient for optimization on Stiefel manifold Edgar Fuentes Figueroa1 · Oscar Dalmau1 Received: 24 September 2019 / Revised: 25 April 2020 / Accepted: 1 May 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we focus on building an optimization scheme over the Stiefel manifold that maintains each iterate feasible. We focus on conjugate gradient methods and compare our scheme to the Riemannian optimization approach. We parametrize the Stiefel manifold using the polar decomposition to build an optimization problem over a vector space, instead of a Riemannian manifold. The result is a conjugate gradient method that averts the use of a vector transport, needed in the Riemannian conjugate gradient method. The performance of our method is tested on a variety of numerical experiments and compared with those of three Riemannian optimization methods. Keywords Steifel manifold · Riemannian optimization · Conjugate gradient · Vector transport · Polar decomposition Mathematics Subject Classification 65K05 · 90C30 · 90C48 · 58C99
1 Introduction In this paper, we consider the following optimization problem with orthogonality constraints: min F (X ) s.t. X X = I p ,
(1)
X ∈Rn× p
where F : Rn× p → R is a differentiable function and I p ∈ R p× p represents the identity matrix. The feasible set St(n, p) = {X ∈ Rn× p |X X = I p }, referred to as the Stiefel manifold, is simplified to the unit sphere when p = 1 and in the case p = n is the Orthogonal Group, O(n). The Stiefel manifold is an embedded submanifold of the vector space Rn× p with dimension equal to np − 21 p( p + 1) Absil et al. (2009). Problem (1) attracts attention
Communicated by Joerg Fliege.
B
Oscar Dalmau [email protected] Edgar Fuentes Figueroa [email protected]
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Center for Research in Mathematics, CIMAT, Guanajuato, Mexico 0123456789().: V,-vol
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E. F. Figueroa, O. Dalmau
of researchers due to its many applications such as the linear eigenvalue problem (Golub and van Loan 2013; Saad 1992; Wen et al. 2016), the orthogonal procrustes problem (Eldén and Park 1999; Schönemann 1966; Gower and Dijksterhuis 2004), the weighted orthogonal procrustes problem Francisco and Martini (2014), heterogeneous quadratic minimization Balogh et al. (2004), the joint diagonalization problem (Joho and Mathis 2002; Theis et al. 2009), the nearest low-rank correlation matrix problem (Grubiši´c and Pietersz 2007; Pietersz and Groenen 2004; Li and Qi 2011; Zhu 2015), Kohn–Sham total energy minimization (Yang et al. 2009; Liu et al. 2015), among others. Riemannian optimization, or optimization over manifolds, considers the Riemannian structure of feasible sets to derive efficient constraint-preserving iterative schemes. In this regard, classical methods for unconstrained optimization Nocedal and Wright (2006) are generalized for optimization on Riemannian manifolds (Abrudan et al. 2008; Nishimori and Akaho 2005; Manton 2002; Zhu 2017; Abrudan et al. 2009; Sato and Iwai 2015;
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