On matrix exponentials and their approximations related to optimization on the Stiefel manifold

被引：6

作者：

Zhu, Xiaojing ^{[1
]}

Duan, Chunyan ^{[2
,3
]}

机构：

[1] Shanghai Univ Elect Power, Coll Math & Phys, Shanghai 200090, Peoples R China

[2] Tongji Univ, Sch Econ & Management, Shanghai 200092, Peoples R China

[3] Univ Arkansas, Dept Ind Engn, Fayetteville, AR 72701 USA

来源：

OPTIMIZATION LETTERS | 2019年 / 13卷 / 05期

基金：

中国国家自然科学基金;

关键词：

Riemannian optimization; Stiefel manifold; Geodesic; Retraction; Matrix exponential;

D O I：

10.1007/s11590-018-1341-z

中图分类号：

C93 [管理学]; O22 [运筹学];

学科分类号：

070105 ; 12 ; 1201 ; 1202 ; 120202 ;

摘要：

In this paper we probe into the geometric properties of matrix exponentials and their approximations related to optimization on the Stiefel manifold. The relation between matrix exponentials and geodesics or retractions on the Stiefel manifold is discussed. Diagonal Pade approximation to matrix exponentials is used to construct new retractions. A Krylov subspace implementation of the new retractions is also considered for the orthogonal group and the Stiefel manifold with a very high rank.

引用

页码：1069 / 1083

页数：15

共 22 条

[1] PROJECTION-LIKE RETRACTIONS ON MATRIX MANIFOLDS [J].

Absil, P. -A. ;

Malick, Jerome .

SIAM JOURNAL ON OPTIMIZATION, 2012, 22 (01) :135-158

[2]

Absil PA, 2008, OPTIMIZATION ALGORITHMS ON MATRIX MANIFOLDS, P1

[3]

[Anonymous], 2013, Matrix Analysis

[4]

[Anonymous], 1975, Essentials of Pade Approximants

[5]

[Anonymous], 2008, Functions of matrices

[6] 2-POINT STEP SIZE GRADIENT METHODS [J].

BARZILAI, J ;

BORWEIN, JM .

IMA JOURNAL OF NUMERICAL ANALYSIS, 1988, 8 (01) :141-148

[7]

do Carmo M. P., 1976, DIFFERENTIAL GEOMETR

[8] The geometry of algorithms with orthogonality constraints [J].

Edelman, A ;

Arias, TA ;

Smith, ST .

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 1998, 20 (02) :303-353

[9] HIGH-ORDER RETRACTIONS ON MATRIX MANIFOLDS USING PROJECTED POLYNOMIALS [J].

Gawlik, Evan S. ;

Leok, Melvin .

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 2018, 39 (02) :801-828

[10] The scaling and squaring method for the matrix exponential revisited [J].

Higham, NJ .

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 2005, 26 (04) :1179-1193

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